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
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
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.