Question

How to solve?
\(x"'+x'(x^2-2)=0\)
Please, help

Answer 1

Is that a third derivative? x''' ?

Answer 2

Yes,

Answer 3

Thanks anyway, I think I got it.

Answer 4

What did you get?

Answer 5

Actually, it comes from traveling wave equation, which is
u(x,t) = f(x-ct)
find the solution for \(u_t+u^2u_x+u_{xxx}=0\)

Answer 6

This is what I have
\(u_t= -cf'\\u_x=f'\\u_{xxx}=f"'\)
Combine all, I have \(f"' =cf' -f^2f'\)
Integral both sides I have

Answer 7

\(f" = cf -\dfrac{f^3}{3}\)

Answer 8

multiple both sides by f', then take integral both sides again, I have
\(\dfrac{f'^2}{2}=\dfrac{cf^2}{2}-\dfrac{f^4}{12}\)

Answer 9

Simplify and factor f^2 out, transfer to the LHS
\((\dfrac{f'}{f})^2=c-f^2/6\)

Answer 10

integral both sides again,
\(ln f = cf -f^3/18\)

Answer 11

so, from here, just solve cubic polynomial.