1. Can someone confirm my working out please

The question comes something like this...

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

2. I'm not sure, because my first equation is non linear?
Why do you say that?

3. I never liked non linear equations to be honest, they seem to always bring problems... hence getting it wrong.

But I feel everything I've done seems ok....

4. I'm saying I don't think your equation is nonlinear. Why do you think it is?

5. Hmmm.. I see, I interpreted B''''E as x*y, they are both a function of x.

6. 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.