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This is not a homework question, it was a question on my test a few days ago. I could not solve it.

Out of memory, the problem was a rod of length L with an end fixed in a wall and the other end free. Its motion satisfies the PDE ##a^4\frac{\partial ^4 u }{\partial x^4} + \frac{\partial ^2 u}{\partial t^2}=0## where ##a>0##.

The boundary conditions are ##u(0,t)=\frac{\partial u}{\partial x}(0,t)=0## and ##\frac{\partial ^2 u}{\partial x^2} (L,t)=\frac{\partial ^3 u}{\partial x^3 }(L,t)=0##.

I had to show that its eigenfrequencies satisfy the following relation: ##\cosh \left ( \frac{\sqrt \omega L }{a} \right ) = \sec \left ( \frac{\sqrt \omega L }{a} \right )##.

Attempt at solution: First off the problem struck me like a hammer since there was no similar problem in my assignments.

I tried to solve the PDE usng separation of variables, until the professor told us "there's no need to solve the PDE. Of course you can do it but it's extra work" which basically means to me that there's some trick I totally missed.

Anyway what I had reached after separation of variable ##u(x,t)=X(x)T(t)## is ##T(t)=A\cos (\lambda t )+B\sin (\lambda t)## where lambda is a constant of separation.

The remaining ODE I had to solve was ##a^4 \frac{X''''}{X}-\lambda ^2 =0##. I assumed the solution was of the form ##X(x)=Ae^{kx}+B^{-kx}## (now I realize it's wrong since it's of order 4 so there should be 4 linearly independent terms, not 2) and I reached after plugging it back into the ODE that ##k=\pm \frac{\lambda }{a}## which gave me ##X(x)##. I tried a few other things like trying to find out the constants of the general solution to the PDE (but I've got X(x) wrong since it's of order 4, not 2) but I reached nothing. Obviously I missed a trick.

So I don't really know how to proceed to solve the problem, even at home. Has anyone an idea?

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# Fourth order PDE

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