Given that for some function $u(x,t)$, if $u_{txt} = 4x$, then $u(x,t) = x^2 t^2 + A(t) + B(x)$.
So, I integrate backwards and seem to get stuck:
- I first integrate with respect to t, and get: $4xt + B(x)$.
- Then, I integrate with respect to x, and get: $2x^2 t + b(x)$, where $b(x) = \int B(x) dx$.
- Now, I integrate with respect to t again, and get: $x^2 t^2 + A(x) + \int b(x) dt$, which is $x^2 t^2 + A(x) + t b(x)$ plus another constant. I don't know how to resolve the $t b(x)$ term, but everything else matches.