general solution for a 4th order PDE I have a fourth order partial differential equation of motion of a tube, with clamped boundary conditions, I don't know what would be the general solution for $W$: 
$$EI \frac{d^4 w(x,t)}{dx^4} + MU^2 \frac{d^2 w(x,t)}{dx^2} + 2MU\frac{d^2 w(x,t)}{dx\,dt} +M \frac{d^2 w(x,t)}{dt^2}=0$$
I need to know the general solution (mode shape) for $w$ (displacement).
$M, E,I,U$ all are known and constant ($U$ is the velocity of a fluid inside the tube).
 A: If you want a single Fourier mode, you're looking for a solution of the form
$$W=e^{ikx+\omega t}$$
The only thing you're missing is the dispersion function $\omega(k)$. If you plug the above solution into your equation, you get the condition
$$EI k^4-M (k U-i \omega )^2=0\ ,$$
which has the solution
$$\omega = i \left(\pm\sqrt{\frac{EI}{M}} k^2 -  U k\right)$$
The general solution is a (possibly infinite) superposition of modes which satisfy this relation. The clamped boundary conditions give you a constraint on the possible values of $k$. 
A: if i use separation of variables [ w=S(x)*G(t) ] then i would have this Eq.: 
A1 * (d^4 S/dx^4) + A2 * (d^2 S/dx^2) + lambda * S =0
in which A1 & A2 are known constants from E,I,M,...
in this case the term (d^2 w/dsdt) would be dropped down (eliminate)
then i have the roots of characteristic equation this way :
q1= (-A2/(2A1)) + [ sqrt(A2^2 - 4*lambda*A1) ]/(2A1)
q2= (-A2/(2A1)) - [ sqrt(A2^2 - 4*lambda*A1) ]/(2A1)
then my general solution would become:
S(x)=C1 sin [(sqrt(q1))*x] + C2 cos [(sqrt(q1))*x] + C3 sinh [(sqrt(q2))*x] + C4 cosh [(sqrt(q2))*x]
where C1,...,C4 are unknown constants to be determined by boundary conditions.
is my procedure correct ??
