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How does one solve a fourth-order PDE of the form $\frac{\partial^4y}{\partial x^4}=c^2\frac{\partial^2y}{\partial t^2}$? It looks like a one dimensional wave equation, but I'm unfortunately very bad at PDEs.

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Well, you can use the good old method of se –  Chris Dugale Sep 11 '12 at 4:10
As Chris was saying, just suppose $y(x,t)=X(x)T(t)$ and subsitute that into the problem to obtain $X''''T=c^2XT''$. Divide by $XT$ etc... –  James S. Cook Sep 11 '12 at 4:17

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up vote 3 down vote accepted

You do almost the same thing as people explained in your other question. Unfortunately, you can only factor the operator into

$$ \left(\frac{\partial^2}{\partial x^2} - c\frac{\partial}{\partial t}\right) \left(\frac{\partial^2}{\partial x^2} + c\frac{\partial}{\partial t}\right)y = 0. $$

Then you have to solve a heat-equation like equation.

If your domain is finite, you should try separation of variables.

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Why does the finitiness of the domain matter for the separation of variables? –  Lurco Jul 21 at 13:35

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