# Is this non-linear equation solvable?

• November 2nd 2012, 01:38 AM
isatis55
Is this non-linear equation solvable?
Hi,I need to solve this equationf''(x) + a*f'(x)^2 +b* f'(x)=0.Any clue?Thanks a lot,Isatis
• November 2nd 2012, 02:20 AM
chiro
Re: Is this non-linear equation solvable?
Hey isatis55.

Consider u = f'(x), then:

du/dx + a*u^2 + b*u = 0 or
du/dx = -a*u^2 - b*u which is separable.
• November 2nd 2012, 04:42 AM
TheEmptySet
Re: Is this non-linear equation solvable?
Quote:

Originally Posted by isatis55
Hi,I need to solve this equationf''(x) + a*f'(x)^2 +b* f'(x)=0.Any clue?Thanks a lot,Isatis

First to simplify the notation I will use $f(x)=y$

While harder for this problem, in general, any 2nd order ODE can be reduced to a first order ODE by the substitution $u=y'$

By the chain rule we get

$\frac{d}{dx}y'=\frac{du}{dy}\underbrace{\frac{dy}{ dx}}_{y'=u} \iff y''=u\frac{du}{dy}$

This will change $y$ from the dependent variable to the independent variable of a new function u.

This gives

$u\frac{du}{dy}+au^2+bu=0 \iff \frac{du}{dy}+au=-b$

This can be solved by an integrating factor.