So I first want to find the general solution for:
$2U_t+U_{xt}=0$ where $U=U(x,t)$
Once I have a general solution, I also want to find the solution for the following initial value problem:
$U(x,0)=0$ , $U_t(x,0)=e^{2x}$
So far, my best attempt has been as follows:
1) Integrate both sides with respect to $t$ to get: $2U+U_x=f(x)$
2) Using integrating factor method:
$\partial/\partial x (Ue^{2x})=f(x)e^{2x}$
3) Integrate and solve for $U(x,t)$:
$U(x,t)=e^{-2x}\int f(x)e^{2x}dx$
Assuming everything I did here is correct, this seems to imply that $U$ is only a function of $x$.
But, this would disagree with the initial value problem, since $U_t(x,0)=e^{2x}$.
Am I missing something here?
Any advice would be greatly appreciated. Thank you in advance.