# Third order nonlinear-ODE

1. Dec 19, 2006

### homeros_81

1. The problem statement, all variables and given/known data
I'm trying to solve a boundary layer equation but i dont really know how. The same problem can be found in Kundu's book 'Fluid Mechanics' there the answer is just written out, but he mentions that it is solved by closed form.

2. Relevant equations
The equation looks like this:
f'''+(1-f'^2)=0

3. The attempt at a solution
This is how far i have got:
Multiplicate with f''
f''f'''+f''-f''f'^2=0
d/ds(f''^2/2)+d/ds(f')-d/ds(f'^3/3)=0
Integrate
f''^2/2+f'-f'^3/3=C
Let g=f'
g'=f''
g'^2+2g-2g^3/3=D
g'=sqrt(2g^3/3-2g+D)
dg/ds=sqrt(2g^3/3-2g+D)
Separable
1/sqrt(2g^3/3-2g+D)*dg=ds

Putting this in to maple gives a really complex expression, there i have no idea how to solve for g.
Does someone have any idea how to do this?

2. Dec 19, 2006

### dextercioby

It's an elliptic integral. It looks nasty, indeed, but that's what the solution is.

If y=f'(x), then

$$x+\bar{C}=\int \frac{dy}{\sqrt{C-2y-\frac{2}{3}y^{3}}}$$

Daniel.