I have an the equation $2U_t + U_{xt} = 0$. I found the general solution as $$U(x,t)=e^{-2x}P(t)+Q(x)$$ I have two questions about this:
1-find the solution for $U(x,0)=0$; $U_t(x,0)=e^{-2x}$
2-Is there a solution for $U(x,0)=0$; $U_t(x,0)=1$
I will appreciate any suggestion.
For your first question, imposing the general solution in your boundary condition gives $$ e^{-2x}P(0)+Q(x)=0\\e^{-2x}P'(0)=e^{-2x} $$ then any $P, Q$ satisfying $ P'(0)=1, Q(x)=-P(0)e^{-2x} $ gives a solution.
For the second question, I don't think there exist a solution. If imposing your general solution to the second boundary condition, it gives $ P'(0)=e^{2x} $, which means that $P$ has to be a function of $x$. That's a contradiction.