I'm having a hard time trying to solve this DE for x(t) using fourier transform:

$\displaystyle \frac{d^2x(t)}{dt^2}-a^2x(t)=f(t), 0\leq t < \infty$

with $\displaystyle \frac{dx(0)}{dt}=b, x(\infty ) < \infty$

Show that $\displaystyle x(t)=-\frac{b}{a}\exp(-at)-\frac{1}{2a} \int_0^{\infty} dt'f(t')[\exp(-a(t+t'))+\exp(-a|t-t'|)]$

Ty