MAT 275 TEST 2 PRACTICE PROBLEMS
I. The Wronskian.
1. Determine the longest interval in which the given initial value problem is certain to have a unique
twice-differentiable solution.
a) 𝑡(𝑡−4 +3𝑡𝑦 6 6
)𝑦′′ ′ + 4𝑦 = 2, 𝑦( ) = 0, 𝑦′( ) = −1
b) (𝑡+1 +𝑡𝑦 0 0
)𝑦′′ ′ + 𝑦 = sec𝑡, 𝑦( ) = 2,𝑦′( ) = −1
c) (𝑡−4 +3𝑡𝑦 1 1
)𝑦′′ ′ +ln(𝑡)𝑦 = sin𝑡, 𝑦( ) = −2,𝑦′( ) = −1
2. Find the Wronskian of the following pair of functions, .
{3𝑒2𝑡, 𝑡𝑒2𝑡}
3. Consider the ODE 𝑡2𝑦′′ +3𝑡𝑦′+ 𝑦 = 0 with the initial conditions .𝑦(1 1
)= 1, 𝑦′( ) = 1
(i) What is the maximum interval of validity, I, of the solution?
(ii) Verify that the functions 𝑦1(𝑡) = 𝑡−1 and 𝑦2(𝑡)= 𝑡−1ln (𝑡) satisfy the ODE for 𝑡 in the interval I.
(iii) Find the Wronskian 𝑊(𝑦1,𝑦2) to show that 𝑦1(𝑡) 𝑦 and 2(𝑡) form a fundamental set of solutions.
(iv) Solve the initial value problem.
4. Suppose 𝑦1(𝑡)= 𝑡 and 𝑦2(𝑡)= 𝑡2 are both solutions of the second order linear equation
𝑦 𝑦
′′ +𝑝(𝑡)′+𝑞(𝑡)= 0. Which of the functions below are guaranteed to also be solutions of the same
equation?
A. 𝑦 = 𝑡2−1 B. C.𝑦 = 5𝑡 𝑦 = −9𝑡2+17𝑡 D. 𝑦 = 0
5. Which of the following is NOT a fundamental set of solutions for ?𝑦′′ −𝑦 = 0
A. {𝑒𝑡, 𝑒 , 2𝑒
−𝑡}B. {2𝑒𝑡 −𝑡}C. {𝑡𝑒𝑡, 𝑒−𝑡} D. {(𝑒𝑡+𝑒−𝑡),1
2(𝑒𝑡+𝑒−𝑡)}
E. {1
2(𝑒𝑡+𝑒−𝑡),1
2(𝑒𝑡−𝑒−𝑡)}F.{1
2(𝑒𝑡+𝑒−𝑡), 𝑒𝑡}
II. HODEs/IVP with constant coefficients.
1. Find a real valued solution to the following initial value problems.
a. 𝑦′′ −6𝑦 0 0
′+13𝑦 = 0, with 𝑦( ) = 1, 𝑦′( ) = 1.
b. 𝑦′′ +4𝑦 0 0
′+4𝑦 = 0, with 𝑦( ) = 1, 𝑦′( ) = −4.
c. 6𝑦 +7𝑦 0 0
′′ ′ +2𝑦 = 0, with 𝑦( ) = 7, 𝑦′( ) = −4.
III. Reduction of order:
1. The ODE has a solution 𝑡2𝑦′′ +3𝑡𝑦′+𝑦 = 0 𝑦1(𝑡)=1𝑡 for . Find the general solution. 𝑡 > 0
2. The ODE 2𝑡𝑦 −5𝑦
′′ ′ +3𝑡𝑦 = 0 has a solution 𝑦1(𝑡)= 𝑡3 for . Find the general solution.𝑡 > 0
IV. Undetermined coefficients
1. Find a particular solution and the general solution of the ODE . 𝑦′′ +2𝑦 +5𝑒
′+𝑦 = 3𝑡2 2𝑡
2. Find a particular solution and the general solution of the ODE: .𝑦′′ −𝑦 3𝑡
′−2𝑦 = 4sin( )
3. Find a particular solution and the general solution of the ODE: .𝑦′′ −𝑦′−12𝑦 = 3𝑡𝑒2𝑡
4. Determine a suitable form for the particular solution , if the method of undetermined𝑌(𝑡)
coefficients is to be used. You do not need to determine the values of the coefficients.
(i) 𝑦′′ +3𝑦 +𝑡 +sin 3𝑡
′= 2𝑡2 2𝑒−3𝑡 ( )
(ii) 𝑦′′ +𝑦 = 𝑡(1+sin𝑡)
(iii) 𝑦′′ −5𝑦 2𝑡 3𝑡+4 sin
′+6𝑦 = 𝑒𝑡cos( )+( )𝑒2𝑡 (𝑡)
(iv) 𝑦′′ +2𝑦 +2𝑒 +4𝑡 sin
′+2𝑦 = 3𝑒−𝑡 −𝑡 cos(𝑡)2𝑒−𝑡 (𝑡)
(v) 𝑦′′ −4𝑦 +4𝑡𝑒 +𝑡sin 2𝑡
′+4𝑦 = 2𝑡2 2𝑡 ( )
V. Mass-Spring system
1. Solve for the position (in meters) of a mass attached to a spring with damping if the mass is no 𝑚 =
1kg, the spring constant is 𝑘 = 4𝑁
𝑚, and and 𝑥(0)= −3m 𝑥′(0)= 6𝑚
𝑠. Also write your answer in
𝐴cos(𝜔𝑡−𝛼) form.
2. Solve for the position (in meters) of a mass attached to a spring with damping if the mass is kg,𝑚 = 3
the damping constant is 𝑐 = 2𝑁∙𝑠
𝑚, the spring constant is 𝑘 = 37
3𝑁
