Consider the problem:
$$ \begin{cases} u_{tt} = c^2 u_{xx} + h(x,t), \hspace{0.5cm} x \in \mathbb{R}, \hspace{0.3cm} t>0\\ u(x,0) = f(x), \hspace{0.5cm} x \in \mathbb{R}\\ u_{t}(x,0) = g(x) \hspace{0.5cm} x \in \mathbb{R} \end{cases} $$
I want to use Fourier transform to formally find a solution to this problem. I solved the homogeneous case in which $h(x,t) = 0$, then I wanted to proceed analogously with this case, but the problem leads me to a nonhomogeneous second-order ordinary differential equation. This is:
Applying Fourier transform on the variable $x$, the following is obtained:
$$\mathcal{F}\{ \partial_{tt}(x,t)\} = c^2 \mathcal{F}\{\partial_{xx}u(x,t)\} + \mathcal{F}\{h(x,t)\} \Rightarrow \partial_{tt}\hat{u} = -c^2 \omega\hat{u} + \hat{h}(\omega,t)$$
How can I continue with this? Or is my way of doing this wrong and there is a better way to do it instead? Any help is appreciated