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I completed the separation of variables step but I am very confused on how to apply the initial any boundary conditions to solve the problem. Please give me some advice or help to go about solving this problem, thank you!

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Problem 1: Solution to homogeneous PDEs by separation of variables
a. Apply separation of variables to derive the general solution $u(x,t)$ to the following homogeneous PDE over the interval $0\leq x\leq 1$, with boundary conditions $u(0,t) = u(1,t) = 0$: $$u = c\frac{\partial^3u}{\partial t\,\partial x^2}$$ b. Find the particular solution with initial conditions $u(x,0)=1$ for $0<x<1$.

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    $\begingroup$ The boundary conditions arise in selecting the eigenfunctions. The initial conditions determine the "Fourier coefficients" of the eigenfunctions in the expansion. $\endgroup$
    – Ian
    Sep 30, 2016 at 21:47
  • $\begingroup$ Formatting tips here. $\endgroup$
    – Em.
    Sep 30, 2016 at 23:32

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Since you did the separation of variables, you have something like $u(x,t) = a(x)b(t)$. The condition that $u(0,t) = 0$ means that $a(0)b(t) =0$. So you must have $a(0)=0$, because if you set $b(t) = 0$ it forces $u(x,t) =0$ and we have no use for the trivial solution. Likewise the second condition gives us $a(1) =0$. Similarly the initial condition says $b(0) =1$.

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