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(4) − 12𝑦’’’ + 53𝑦’’ − 100𝑦’ + 68𝑦 = 500𝑒2x b) 𝑥3𝑦’’’− 7𝑥2𝑦’’ + 24𝑥𝑦’ − 34𝑦 = 50𝑥2, 𝑥 > 0 c) 8𝑦’’’ − 12𝑦’’ + 6𝑦’ − 𝑦 = 960𝑒x⁄2 d) 𝑥2𝑦’’ = 𝑦′(3𝑥 − 2𝑦’) e) 4𝑦(𝑦’)2𝑦’’ = (𝑦’)4 + 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:
2𝑦3 − 4𝑒-x𝑦’ – 𝑦’’ = 𝑥2 sin 𝑥 , 𝑦(0) = −3, 𝑦’(0) = 1