How can I solve a third order nonlinear ODE for a boundary layer equation?

[homeros_81]

Homework Statement

I'm trying to solve a boundary layer equation but i don't 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.

Homework Equations

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

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 into 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?

 

[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.