Why do you say that?I'm not sure, because my first equation is non linear?
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.
Well, even in looking at the original pde: it's linear in u. u and none of its derivatives appear to anything except the first power, and there are no functions of u or any of its derivatives. It's a linear pde.
I think everything in the OP is fine, until you get to the separation step. The parameters I and A should only appear in one or the other of the B and C ordinary differential equations. Also, double-check what's in the numerators and the denominators of those ode's.
Incidentally, a pde being nonlinear does not imply that separation of variables won't work. It probably won't work. Dym's equation is a (rare) nonlinear pde that succumbs to separation of variables.