Third Order Non-Linear Homogeneous ODE

[Compressible]

I have derived a 3rd order non-linear ODE with its respective boundary conditions, and I was hoping to get a hint on how to find a closed form solution to it. The equation is given as:

F''' + (1/C^2)*F*F' = 0

Where the primes denote a derivative, and C is just a constant. Any help is highly appreciated!

 

[Mark44]

I have derived a 3rd order non-linear ODE with its respective boundary conditions, and I was hoping to get a hint on how to find a closed form solution to it. The equation is given as:

F''' + (1/C^2)*F*F' = 0

Where the primes denote a derivative, and C is just a constant. Any help is highly appreciated!

I think this might help. Write the equation as F''' = -(1/C²)FF'
Now integrate both sides. That will get you a 2nd order DE.

 

[Compressible]

So I tried what you mentioned, and here's what I get (though I think I made a mistake somewhere):

F''' = -(1/C^2)*F*F'
> F'' = -(1/2/C^2)*F^2 + C1 (where C1 is a new constant)

Now, this gives:

F'' + (1/2/C^2)*F^2 + C1 = 0

Assume G = F'
F'' = G*dG/dF

> G*dG/dF + (1/2/C^2)*F^2 + C1 = 0
> G*dG = [-(1/2/C^2)*F^2 - C1]*dF

Integrating both sides gives,

G = sqrt(-(1/3/C^2)*F^3 - 2*C1*F + 2*C2) = F' = dF/dy

Separating variables gives,

dF/sqrt(-(1/3/C^2)*F^3 - 2*C1*F + 2*C2) = dy

But now I have no idea how to take that integral and I'm pretty sure I made some mistakes with the constants. Any suggestions as to where I went wrong and what I can alter?

 

[jackmell]

You said closed-form right? First just write it as:

\frac{dy}{dx}=\sqrt{ay^3+by+c}

Now, suppose I told you there is a special function called the Weierstrass P function such that if:

\text{myxside}=\int_{\infty}^{y}\frac{dt}{\sqrt{4t^3-ct-d}}

then:

y=P(\text{myxside,c,d})

and you just had to make some kind of showing on this thing, could you express your solution in terms of the Weierstrass P function?

 

[Compressible]

You said closed-form right? First just write it as:

\frac{dy}{dx}=\sqrt{ay^3+by+c}

Now, suppose I told you there is a special function called the Weierstrass P function such that if:

\text{myxside}=\int_{\infty}^{y}\frac{dt}{\sqrt{4t^3-ct-d}}

then:

y=P(\text{myxside,c,d})

and you just had to make some kind of showing on this thing, could you express your solution in terms of the Weierstrass P function?

Honestly, I'm not familiar with the Weierstrass P function. I have to solve this ODE and apply three boundary conditions to it. I'm not sure how the Weierstrass P function operates to be able to do that.

 

[Compressible]

Looking at it again, I think I found a way to solve the ODE, if the 1/C^2 constant wasn't there. Is there any way to define a new variable F that absorbs that constant value. If I can find a way to do that, then that ODE is easily solvable.

 

[Compressible]

Nevermind. Solved! Thanks for the help!