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

[roldy]

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) = u_xx(0,t) = u_xx(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.

 

[HallsofIvy]

No, there is no "simple way to do this just by observation".

 

[roldy]

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.