## Determining wheter or not a non trivial solutions exists for higher order PDE's

Is there a way to determine if a non trivial solution exists for higher order PDE's? For example suppose I have the following.

$X''''(x) + \alpha^2X(x)=0$

With given conditions U(0,t) = u(1,t) = uxx(0,t) = uxx(1,t) = 0 if t≥0

The general solution will have 4 constants of which I will have to solve for using the above conditions. However, if this shows up on a test I will not have enough time to do this and the other parts that go along with this step. There's got to be a simple way to do this just by observation.

Note: This step is part of a process to solve the beam equation. I've done the case with λ=0 and λ<0. The case for λ>0 is where I need to find a shorter faster way. I have tried searching online for solution examples to solving this but all the solutions just jump to the case where λ<0.
 PhysOrg.com science news on PhysOrg.com >> King Richard III found in 'untidy lozenge-shaped grave'>> Google Drive sports new view and scan enhancements>> Researcher admits mistakes in stem cell study
 Recognitions: Gold Member Science Advisor Staff Emeritus No, there is no "simple way to do this just by observation".
 Is there any other way without going through the process of figuring out the general solution? My professor hinted that there was but I can't figure it out. Perhaps because this is a beam, the eigenvalues have to be real due to the fact that they represent the frequencies.