I'm trying to solve a variant of the wave equation IVP. I believe the solution might involve completing the square, but I have not found a method that works. Given the following, does anyone have a solution (with proof) for the problem:
$$ u_{tt} - 10u_{xt} + 3u_{xx} = 0 \\ u(x,0) = e^{-x^{2}} \\ u_{t}(x,0) = 2\sin (x) $$
$$ (\partial_t^2-10\partial_x\partial_t+3\partial_x^2)u=0 \\ (\partial_t-5\partial_x)^2u-22\partial_x^2u=0 \\ (\partial_t-(5+\sqrt{22})\partial_x)(\partial_t-(5-\sqrt{22})\partial_x)u=0 $$ Therefore, \begin{align} u(x,t)&=f((5+\sqrt{22})t+x)+g((5-\sqrt{22})t+x) \\ u(x,0)&=f(x)+g(x) = e^{-x^2} \\ u_t(x,0)&=(5+\sqrt{22})f(x)+(5-\sqrt{22})g(x) \\ &= 5e^{-x^2}+\sqrt{22}(f(x)-g(x)) = 2\sin(x) \end{align} So $f+g=e^{-x^2}$ and $f-g=\frac{1}{\sqrt{22}}(2\sin(x)-e^{x^2})$, which determines $f$ and $g$ and also the solution $u(x,t)$.