Non-linear Differential Equation - Pulling my Hair

[jkent]

Non-linear Differential Equation - Pulling my Hair !

Hi,
What seems like a simple problem could be going abit better. Any ideas would be sincerely appreciated.

(y'')^2 -xy'' + y' = 0
The squared term is causing me grief !
If I set say v = y' , that still leaves me with the squared term.
(v')^2 - xv' + v =0.

Sorry .. I must be missing something. This looks remarkably quadratic - but its not triggering anything for me right now.

Ideas please and thank you !

J. Kent.

 

[jackmell]

When you let v=y', then we get:

$$v=xv'+f(v')$$

That's a common non-linear equation. Wanna' look for it?

 

[Eynstone]

Try substituting x=rcost, 4v=(rsint)^2.

 

[JJacquelin]

That's a common non-linear equation

A clue : a Clairaut's ODE
Finally:
y = a x²-4a²x+b
a , b = constants

 

[jkent]

Good ideas all. I'll poke at this abit more. Claurauts ODE - hasn't made my list so far - but it clearly needs to. I was told that factoring will work. I just don't remember enough of this stuff and use it infrequently ! Thank you so much !