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Study Guide for Exam 3

Show absolutely all work. Start each problem on a new page. Label them clearly.

The Problems

1. Find the general solutions of the given differential equations, initial value problem, and system using whichever method that seems appropriate.

a) 𝑦(") − 12𝑦$$$ + 53𝑦$$ − 100𝑦$ + 68𝑦 = 500𝑒%&

b) 𝑥’𝑦$$$ − 7𝑥%𝑦$$ + 24𝑥𝑦$ − 34𝑦 = 50𝑥%, 𝑥 > 0

c) 8𝑦$$$ − 12𝑦$$ + 6𝑦$ − 𝑦 = 960𝑒&⁄%

d) 𝑥%𝑦$$ = 𝑦′(3𝑥 − 2𝑦$)

e) 4𝑦(𝑦$)%𝑦$$ = (𝑦$)" + 3, assume x is the independent variable.

f) 2𝑥$ + 2𝑥 + 𝑦$ − 𝑦 = 𝑡 + 1,𝑥$ + 3𝑥 + 𝑦$ + 𝑦 = 4𝑡 + 14. (Eliminate x first!)

2. Find the first six non-zero terms of a Taylor series solution to the initial value problem:

2𝑦’ − 4𝑒)&𝑦$ − 𝑦$$ = 𝑥% sin 𝑥 ,𝑦(0) = −3,𝑦$(0) = 1

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