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We have the equation $u_{tt}= u_{xx}$ for $0<x<2 $ and $t>0$ subject to $u(x,0)= 0$, $u_t(x,0) = x$, $u(0,t)=t$, and $u_x(2,t) =t^2$. Using variable separation method, we set $u(x,y)=X(x)T(t)$. We get T(0)=0 from the first condition. I am not able to conclude anything from other options. Is it better to use Fourier transforms for these kind of problems?

User2018
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  • The problem is overdetermined. The two conditions $u(x,0)=0$ and $(\partial_t u)(x,0)=x$ already uniquely specify a solution. – K.defaoite Dec 04 '21 at 11:32
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    If you were going to use separation of variables, you need to make the boundary conditions $u(0,t) = t$ and $u_{x}(2,t) = t^{2}$ homogeneous i.e you need them to be of the form $u(0,t) = 0$ and $u_{x}(2,t) = 0$. If you want to make your boundary conditions homogeneous, let $u(x,t) = v(x,t) + t + xt^{2}$ and now you have the new PDE system in $v$ to solve \begin{align} v_{tt} &= v_{xx} - 2x \ v(0,t) &= 0 \ v_{x}(2,t) &= 0 \ v(x,0) &= 0 \ v_{t}(x,0) &= x-1 \end{align} – Matthew Cassell Dec 04 '21 at 18:02
  • @mattos: I think then we can apply Duhamel’s method – User2018 Dec 05 '21 at 04:23

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