Second order differential equation with Heaviside function

Question

I have a differential equation of the form $$y''(x) - a y(x) + b \theta(c - x) = 0, \quad y(0) = 0, \quad \lim_{x \to \infty} y(x) = 0,$$ where $a$, $b$, $c$ are some constants and $\theta(с - x)$ is the Heaviside function. If there was any condition for the derivative, I would simply use Laplace transform and that's it, but here I seemingly have to search for the solution directly: $$y = C_1 e^{\sqrt a x} + C_2 e^{-\sqrt a x} + C_3 f(x)$$ And I've got no idea what function $f(x)$ might be. Please, help me out.

Answer 1

Oh, it turns out to be pretty easy. The equation simply says that in $x \in [0, c)$ it has the form $y''(x) - a y(x) + b = 0$ and in $x > c$: $y''(x) - a y(x) = 0$. Boundary conditions give trivial solution on $x > c$, so I can treat $\theta$-function on $[0, c)$ as a constant to obtain: $$y(x) = b\theta(c - x)(e^{-\sqrt a x} + 1)$$