The question comes something like this...
The dynamic beam equation (without loading) for displacement
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 separate C(t) and B(x) and then equate everything to a constant say Z, then I get this...
C''=I*Z/A * C
I'm not sure, because my first equation is non linear?
But I feel confident it's right.