Problems 9-7: 7

Also: Solve the following:

1. $y'' - y' = u(x - 1)$, $y(0) = 0$, $y'(0) = 0$
2. $y'' + 3y' + 2y = f(x)$, $y(0) = 0$, $y'(0) = 0$, where $f(x) = \begin{cases}x,& \text{if } 0 < x < 1, \\ 1,& \text{if } x > 1\end{cases}$
3. $y'' + y = g(x)$, $y(0) = 3$, $y'(0) = -1$, where $g(x) = \begin{cases}\cos(x),& \text{if } 0 < x < \frac{\pi}{2}, \\ 0,& \text{if } x > \frac{\pi}{2}\end{cases}$

Answers:

1. $y(x) = (e^{x - 1} - x)u(x - 1)$
2. $y(x) = \frac{x}{2} - \frac{3}{4} + e^{-x} - \frac{1}{4}e^{-2x} + \left(\frac{x - 1}{2} - \frac{3}{4} + e^{-(x - 1)} - \frac{1}{4}e^{-2(x - 1)}\right)u(x - 1)$
3. $y(x) = 3\cos(x) - \sin(x) + \frac{1}{2}x\sin(x) - \frac{1}{2}\left(\left(x - \frac{\pi}{2}\right)\sin(x) + \cos(x)\right)u\left(x - \frac{\pi}{2}\right)$