Help solving non-linear ODE analytically

[member 428835]

Hi PF!

Anyone have any ideas for a solution to this $$0 = F F''+\left.2F'\right.^2+ xF' + F$$ where primes denote derivatives with respect to $x$.
So far I have tried this $$0=\left( F F'\right)'+\left({xF}\right)'+\left.F'\right.^2$$

Which obviously failed. I also thought of this $$0 = F^2 F''+2F\left.F'\right.^2+ xFF' + F^2\\
= (F'F^2)' + (xF^2)'+xFF'$$
which also fails. Any ideas? I know an analytic solution exists, but how to derive it?

 

[Stephen Tashi]

Hi PF!

Anyone have any ideas for a solution to this $$0 = F F''+\left.2F'\right.^2+ xF' + F$$ where primes denote derivatives with respect to $x$.

F = -x is a particular solution. Look up how to do the "reduction of order" of a differential equation. (I say look it up, because I'd have to look it up myself before attempting to explain it.)

 

[member 428835]

F = -x is a particular solution. Look up how to do the "reduction of order" of a differential equation. (I say look it up, because I'd have to look it up myself before attempting to explain it.)

That's an idea, which is all I'm asking for. But $-x$ doesn't solve this. Comes close though.

 

[pasmith]

Hi PF!

Anyone have any ideas for a solution to this $$0 = F F''+\left.2F'\right.^2+ xF' + F$$ where primes denote derivatives with respect to $x$.

I know an analytic solution exists, but how to derive it?

If $F(x) = kx^2$ then every term on the right is a constant times $x$. You can then choose $k$ so that those constants sum to zero.

 

[member 428835]

Thanks! I do know the exact solution is $3(x^{1/3}-x^2)/10$ but was wondering how to find this solution apart from guessing.