differential equation practice test

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MA 238 – Differential Equations – Section 105 – Fall 2017

Model Test 02

50 minutes

Name: …………………………………………………………………………………………………………………………….

Answer all questions in the space provided.

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1. Find the general solutions for the following ODEs.

𝑎) 𝑑3𝑦

𝑑𝑥3 +

𝑑𝑦

𝑑𝑥 = 0

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b) ((𝐷2 + 3)2 (𝐷 − 1)2) 𝑦 = 0 , in which 𝐷 ≡ 𝑑

𝑑𝑥

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2. Find the general solution of the following ODEs. Use undetermined coefficients to find the particular

solution for each ODE.

𝑎) 𝑑2𝑦

𝑑𝑥2 − 9

𝑑𝑦

𝑑𝑥 + 14𝑦 = 𝑥2

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b) 𝑑2𝑦

𝑑𝑥2 − 9

𝑑𝑦

𝑑𝑥 + 14𝑦 = 𝑒 6𝑥

c) 𝑑2𝑦

𝑑𝑥2 − 9

𝑑𝑦

𝑑𝑥 + 14𝑦 = 3𝑥2 + 8𝑒 6𝑥

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3. Solve the following initial value problems.

𝑎) 𝑦′′ + 𝑦 = 4𝑥 + 10 sin(𝑥) , 𝑦(𝜋) = 0, 𝑦′(𝜋) = 2

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𝑏) 4𝑦′′ + 4𝑦′ + 17𝑦 = 0, 𝑦(0) = −1, 𝑦′(0) = 2.

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4. Find the general solution of the following ODE, using variation of parameters to find the

particular solution.

𝑎) 𝑑2𝑦

𝑑𝑥2 − 3

𝑑𝑦

𝑑𝑥 + 2𝑦 = cos(𝑒 −𝑥 )

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𝑏) 𝑑2𝑦

𝑑𝑥2 + 9𝑦 = csc(3𝑥)