If you take the Laplace transform we get
For the first part just evaluate this integral
$\displaystyle \int_{0}^{\infty}e^{at}\cdot e^{-st}dt = \int_{0}^{\infty}e^{-t(s-a)}dt $
$\displaystyle s^2X(s)-s(x(0)-x'(0)-a^2X(s)=F(s) \iff X(s)=\frac{s-1}{xs^2-a^2}+F(s)\cdot \frac{1}{x^2-a^2}$
Now use the convolutions theorem to finish the job remember that
$\displaystyle \mathcal{L}^{-1}\{F(s)G(s) \}=\int_{0}^{t}f(t-u)g(u)du$
Can you finish from here?