Differential Equations MCQ Set Covering Ordinary Differential Equations,
Initial Value Problems, and Applications
1. What is the general solution of the differential equation dy/dx = 2x?
a) y = x^2 + C
b) y = 2x + C
c) y = x^2/2 + C
d) y = 2x^2 + C
Answer: a) y = x^2 + C
Solution:
Integrate both sides of the equation:
∫ dy = ∫ 2x dx
y = x^2 + C
2. Solve the initial value problem: dy/dx = 3x^2, y(0) = 1
a) y = x^3 + 1
b) y = x^3 + 2
c) y = 3x^3 + 1
d) y = x^3/3 + 1
Answer: a) y = x^3 + 1
Solution:
First, integrate: y = x^3 + C
Then, use the initial condition y(0) = 1:
1 = 0^3 + C
C = 1
Therefore, y = x^3 + 1
3. What is the order of the differential equation d^3y/dx^3 + 2(d^2y/dx^2) = sin(x)?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
Solution:
The order of a differential equation is the highest derivative present. Here, the highest derivative is
d^3y/dx^3, which is of order 3.
4. Solve the separable differential equation: dy/dx = y^2 * cos(x)
a) y = -1 / (sin(x) + C)
b) y = 1 / (sin(x) + C)
c) y = -1 / (cos(x) + C)
d) y = 1 / (cos(x) + C)
Answer: b) y = 1 / (sin(x) + C)
Solution:
Separate variables: (1/y^2) dy = cos(x) dx
Integrate both sides: -1/y = sin(x) + C
Solve for y: y = 1 / (sin(x) + C)
5. What type of differential equation is dy/dx + Py = Q, where P and Q are functions of x only?
a) Separable
b) Homogeneous
c) Linear
d) Bernoulli
Answer: c) Linear
Solution:
A linear first-order differential equation has the form dy/dx + P(x)y = Q(x), which matches the given
equation.
6. Solve the homogeneous differential equation: dy/dx = (2x + y) / x
a) y = x(ln|x| + C)
b) y = x(2ln|x| + C)
c) y = 2x + C
d) y = x^2 + C
Answer: b) y = x(2ln|x| + C)
Solution:
Let y = vx, then dy/dx = v + x(dv/dx)
Substitute: v + x(dv/dx) = 2 + v
Simplify: x(dv/dx) = 2
Separate and integrate: ∫(1/2)dv = ∫(1/x)dx
v/2 = ln|x| + C
v = 2ln|x| + 2C
Substitute back y = vx: y = x(2ln|x| + C)
7. What is the integrating factor for the linear differential equation dy/dx + 2y = e^x?
a) e^x
b) e^2x
c) e^-2x
d) e^-x
Answer: b) e^2x
Solution:
For a linear differential equation dy/dx + P(x)y = Q(x), the integrating factor is e^∫P(x)dx.
Here, P(x) = 2, so the integrating factor is e^∫2dx = e^2x.
8. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
. Solve the Bernoulli equation: dy/dx + y = xy^2
a) y = 1 / (x - Ce^x)
b) y = 1 / (Ce^x - x)
c) y = 1 / (x + Ce^x)
d) y = 1 / (Ce^-x + x)
Answer: c) y = 1 / (x + Ce^x)
Solution:
Let v = 1/y, then dy/dx = -v^-2 * dv/dx
Substitute: -v^-2 * dv/dx + v^-1 = xv^-2
Multiply by v^2: -dv/dx + v = x
This is now linear. Solve using integrating factor e^x:
d(ve^x)/dx = xe^x
ve^x = ∫xe^x dx = xe^x - e^x + C
v = x - 1 + Ce^-x
Substitute back y = 1/v: y = 1 / (x + Ce^x - 1)
Absorb the -1 into C: y = 1 / (x + Ce^x)
9. What is the general solution of the second-order linear differential equation d^2y/dx^2 - 4y = 0?
a) y = C1e^2x + C2e^-2x
b) y = C1e^x + C2e^-x
c) y = C1sin(2x) + C2cos(2x)
d) y = C1x^2 + C2x
Answer: a) y = C1e^2x + C2e^-2x
Solution:
The characteristic equation is r^2 - 4 = 0
Solving: r = ±2
The general solution is y = C1e^(r1x) + C2e^(r2x) = C1e^2x + C2e^-2x
10. Solve the initial value problem: d^2y/dx^2 + 9y = 0, y(0) = 2, y'(0) = 3
a) y = 2cos(3x) + sin(3x)
b) y = 2sin(3x) + cos(3x)
c) y = 3cos(3x) + sin(3x)
d) y = cos(3x) + 3sin(3x)
Answer: a) y = 2cos(3x) + sin(3x)
Solution:
General solution: y = C1cos(3x) + C2sin(3x)
Use initial conditions:
y(0) = 2: C1 = 2
y'(0) = 3: -3C1sin(0) + 3C2cos(0) = 3
3C2 = 3, so C2 = 1
Therefore, y = 2cos(3x) + sin(3x)
11. What is the particular solution of dy/dx + 2y = 4x using the method of undetermined coefficients?
a) yp = 2x - 1
b) yp = 2x + 1
c) yp = x - 1
d) yp = x + 1
Answer: a) yp = 2x - 1
Solution:
Assume yp = Ax + B
y'p = A
Substitute into the original equation:
A + 2(Ax + B) = 4x
Equate coefficients:
2A = 4, so A = 2
A + 2B = 0, so B = -1
Therefore, yp = 2x - 1
12. Solve the system of differential equations:
dx/dt = 2x + y
dy/dt = x + 2y
a) x = C1e^3t + C2e^t, y = C1e^3t - C2e^t
b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
c) x = C1e^t + C2e^3t, y = C1e^t - C2e^3t
d) x = C1e^t - C2e^3t, y = C1e^t + C2e^3t
Answer: b) x = C1e^3t - C2e^t, y = C1e^3t + C2e^t
Solution:
Find eigenvalues: |A - λI| = 0
|(2-λ 1)| = 0
|( 1 2-λ)|
(2-λ)^2 - 1 = 0
λ^2 - 4λ + 3 = 0
λ = 3 or 1
For λ = 3: (2-3)x + y = 0, so x = y
For λ = 1: (2-1)x + y = 0, so x = -y
General solution:
x = C1e^3t - C2e^t
y = C1e^3t + C2e^t
13. What is the Wronskian of the functions e^x and e^-x?
a) 0
b) 1
c) 2
d) -2
Answer: c) 2
Solution:
Wronskian W(f,g) = f*g' - f'*g
W(e^x, e^-x) = e^x * (-e^-x) - e^x * e^-x
= -1 - 1 = -2
|W| = 2
14. Solve the Euler-Cauchy equation: x^2(d^2y/dx^2) + 3x(dy/dx) - y = 0
a) y = C1x + C2x^-1
b) y = C1x^2 + C2x^-2
c) y = C1x^1/2 + C2x^-3/2
d) y = C1x^3/2 + C2x^-1/2
Answer: a) y = C1x + C2x^-1
Solution:
Assume y = x^r
Substitute: r(r-1) + 3r - 1 = 0
r^2 + 2r - 1 = 0
(r+1)^2 = 2
r = 1 or -1
General solution: y = C1x^1 + C2x^-1 = C1x + C2x^-1
15. What is the order and degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 = 0?
a) Order: 2, Degree: 3
b) Order: 3, Degree: 2
c) Order: 2, Degree: 2
d) Order: 3, Degree: 3
Answer: a) Order: 2, Degree: 3
Solution:
Order is the highest derivative: d^2y/dx^2 is 2nd order
Degree is the highest power of the highest derivative: (d^2y/dx^2)^3 has degree 3
16. Solve the non-homogeneous differential equation: d^2y/dx^2 + 4y = 8sin(2x)
a) y = C1cos(2x) + C2sin(2x) + sin(2x)
b) y = C1cos(2x) + C2sin(2x) - sin(2x)
c) y = C1cos(2x) + C2sin(2x) + xsin(2x)
d) y = C1cos(2x) + C2sin(2x) - xsin(2x)
Answer: b) y = C1cos(2x) + C2sin(2x) - sin(2x)
Solution:
Complementary solution: yc = C1cos(2x) + C2sin(2x)
Particular solution: Try yp = Asin(2x)
y'p = 2Acos(2x)
y''p = -4Asin(2x)
Substitute: -4Asin(2x) + 4Asin(2x) = 8sin(2x)
A = -1
General solution: y = yc + yp = C1cos(2x) + C2sin(2x) - sin(2x)
17. What is the solution of the initial value problem dy/dx = x^2 + y^2, y(0) = 1?
a) y = tan(x^3/3 + x + π/4)
b) y = tan(x^3/3 + x - π/4)
c) y = cot(x^3/3 + x + π/4)
d) y = cot(x^3/3 + x - π/4)
Answer: a) y = tan(x^3/3 + x + π/4)
Solution:
This is a Riccati equation. Let y = (1/u)(du/dx)
Substitute: (1/u)(du/dx) = x^2 + (1/u^2)(du/dx)^2
u(du/dx) = x^2u^2 + (du/dx)^2
d^2u/dx^2 = x^2u
This is Airy's equation. Its solution is:
u = C1Ai(x) + C2Bi(x)
y = (1/u)(du/dx) = tan(x^3/3 + x + C)
Use initial condition y(0) = 1:
1 = tan(C), so C = π/4
Therefore, y = tan(x^3/3 + x + π/4)
18. Solve the Laplace equation in polar coordinates: (1/r)(∂/∂r)(r∂u/∂r) + (1/r^2)(∂^2u/∂θ^2) = 0
a) u(r,θ) = (A0 + B0ln(r)) + Σ(Anr^n + Bnr^-n)(Cncos(nθ) + Dnsin(nθ))
b) u(r,θ) = (A0 + B0r) + Σ(Anr^n + Bnr^-n)(Cncos(nθ) + Dnsin(nθ))
c) u(r,θ) = (A0 + B0θ) + Σ(Anr^n + Bnr^-n)(Cncos(nθ) + Dnsin(nθ))
d) u(r,θ) = A0 + Σ(Anr^n + Bnr^-n)(Cncos(nθ) + Dnsin(nθ))
Answer: a) u(r,θ) = (A0 + B0ln(r)) + Σ(Anr^n + Bnr^-n)(Cncos
