Solve 4th order PDE using Finite Cosine Fourier Transform to verify solution

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Boundary Conditions Finite Fourier Transform Definition

From the PDE shown above I am trying to solve it using the finite cosine fourier transform.

When applying the finite fourier cosine transform I get the ODE with conditions shown above U_{tttt} + (n^4)*U = 0

however I cannot seem to find a way to solve this, then apply the reverse finite cosine fourier transform and get the solution as shown in the picture above. A method or ideas to find out would be most helpful ​