Given the wave equation: $G_{tt} - G_{xx} = \delta(x-x_{0})\delta(t-t_{0})$,
with conditions: $-\infty < x < \infty,t>0$,$G(x,0) = 0, G_{t}(x,0) = 0$. Use Laplace transform to find the Green's function for this equation.
--
After applying the Laplace transform to the equation in $t$, I get:
$$s^{2}\bar{G} - sG(x,0) - G_{t}(x,0) - \bar{G_{xx}} = e^{-st_{0}}\delta(x-x_{0})$$
By the initial condition:
$$s^{2}\bar{G} - \bar{G_{xx}} = e^{-st_{0}}\delta(x-x_{0})$$
But I am struggling here, after solving the homogenous equation $s^{2}\bar{G} - \bar{G''} = 0$, I got $\bar{G} = A(s)e^{xs}+B(s)e^{-xs}$. And $\bar{G}$ makes the left hand side of the above equation zero. Are there better ways to deal with the left hand side?