The question comes something like this...

The dynamic beam equation (without loading) for displacement

is

Given by d^2/dx^2 (I*E d^2u/dx^2)= A d^2u/dt^2

Where I,A are constant and E is a function of x --> E(x)

Find two ODE's by separation of variable..

So I assumed the equation is in the form u(x,t)=B(x)*C(t)

I find U(xx)= B''(x)C(t) and U(tt)=B(x)C''(t)

Then I differentiate it again, but with E(x), using product rule

I get

I*C(t){B''''(x)E(x)+2B'''(x)E'(x)+B''(x)E''(x)}=A* B(x)*C''(t)

I separate C(t) and B(x) and then equate everything to a constant say Z, then I get this...

B''''E+2B'''E'+B''E''=IZ/A *B

C''=I*Z/A * C

I'm not sure, because my first equation is non linear?

But I feel confident it's right.

Thanks