MATH 361 EXAM 2, Spring 2002

Simplify all answers. Show your work!  Name:  Score
1. a) Solve: 4y′′ + 4y′ + y = 0.  b) Solve: 4y′′′ + 4y′′ + y′ = 0.  1

2 3 4 5

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2. Explain what is meant by the following terms:  7

a) Linear differential equation. b) Principle of superposition.

8 9 10 Tot

3. Let Ly = y′′ + 25y.
a) Solve: Ly = 0.  b) Solve: Ly = 0; y(0) = 3, y′(0) = 0

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4. a) Write in the form a + ib: e^{3+iπ}.  b) Test for l. i.: y_{1} = e^{−3t}; y_{2} = e^{−6t}

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5. Find the Wronskian of two l. i. solutions of the equations:
a) t^{2}y′′ + ty + (25 − t^{2})y = 0  b) (1 + t^{2})y′′ + 2ty′ + 6y = 0

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Part II. Name:
6. Use undetermined coefficients to find the particular solution of y′′ + 4y = sin 3t.

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7. Write a suitable undetermined coefficient form for the particular solution of:

a) y′′ + 4y = 5t^{2} + (2t − 5) cos 2t.

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b) y′′ + 2y′ + 2y = 5te^{2t} + 4e^{−t} cos t

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8. Let y′′ − (4/t)y′ + (6/t^{2})y = 5. It is given that y_{1} = t^{2} is a solution of the homogeneous equation.

a) Use Abel’s formula to find y_{h}.

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b) Use variation of parameters to find y_{p}.

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