deferential equations

Differential Equations MAT239 Fall 2015

Suggested Practice Problems : Section 2.1, 2.2, 2.6 Due Monday September 21, 2015 in class

Section 2.2: Separable ODEs

1. Solve the following ODE.

(a) dy

dx =

e−y 2

2y(1 + x2) , y(1) = 0

(b) y′ = y2 −y x

, y(1) = 2

(c) t2 dy

dt = ln t

(d) y′ = cos2 x √

1 −y2, y(0) = 1 √

2

2. Solve the following homogeneous ODE.

(a) dy

dx = x + 3y

x−y , y(1) = 0

(b) dy

dx =

3y2 −x2

2xy

Section 2.1: First Order Linear ODEs

Solve the following first order linear ODEs.

1. dy

dx −y = 2xe2x, y(0) = 1

2. ty′ + 2y = sin(t2)

3. dy

dx + y cos x = sin x cos x, y(0) = π

Section 2.6: Exact ODEs

1. Determine if the ODE is exact. If it is exact, solve it.

(a) (2xy −y2) + (2xy −x2)y′ = 0 (b) (3x2 − 2xy + 2) dx + (6y2 −x2 + 3) dy = 0 (c) (ex sin y − 2y sin x) + (ex cos x + 2 cos x)y′ = 0

(d) (t ln y + ty) + (y ln t + ty) dy

dt = 0, t > 0, y > 0

2. Verify that the following non-exact ODE becomes exact on multiplication by the IF

µ = 1/xy3: x2y3 + (x + xy2) dy

dx = 0

3. Find an IF µ that makes the following non-exact ODE exact and solve it: y′ = e2x+y−1

Answers

Section 2.2: Separable ODEs

1. (a) y2 = ln |tan−1 x + (1 − π 2 )|

(b) y = 2

2 − cx

(c) y = 1 + ln t

−t ln t + c

(d) y = π + 2x + sin(2x)

4

2. (a) 2x

(x + y) + ln |x + y| = 2 and y = −x

(b) c|x|3 = |y2 −x2|

Section 2.1: First Order Linear ODEs

1. y = 2(x− 1) + 3ex

2. y = − cos(t2)

2t2 + c

t2

3. y = (sin x− 1) + (π + 1)esinx

Section 2.6: Exact ODEs

1. (a) Not exact

(b) x3 −x2y + 2x + 2y3 + 3y = c (c) ex sin y + 2y cos x = c and y = 0

(d) Not exact

2. Show My = Nx

3. µ = e−x and the solution is y = cex + 1 + e2x

2