Solve equation:
$\displaystyle -u_{xxxx}(x,t)=\lambda u(x,t)$, $\displaystyle 0<x<L$
^{$\displaystyle u(0)=u(L)=0 $, $\displaystyle u_{xx}(0)=u_{xx}(L)=0$ }
You don't really need "DSolve". The general solution to any linear homogeneous differential equation is a linear combination of the elementary solutions. The conditions that all values be 0 then must give u identically equal to 0 as the solution. (And it should be easy to see that u= 0 for all x does satisfy the equation and additional conditions.)