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