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Math Help - Is this non-linear equation solvable?

  1. #1
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    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
    Last edited by isatis55; November 2nd 2012 at 01:59 AM.
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  2. #2
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    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.
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    Re: Is this non-linear equation solvable?

    Quote Originally Posted by isatis55 View Post
    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.
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