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I'm trying to solve the non linear differential equation $yy'+xy=x^3$.

Making the substitution $u=\dfrac{x^2}{y}$, solving for $y$ and taking the derivative, gives

$y'=\dfrac{2ux-u'x^2}{u^2}$

With that substitution, the equation turns into a separable one, like this $$u'=\dfrac{-(u^3-u^2-2u)}{x}$$

My question is: Is it correct to make a substitution that is nonlinear for $x$? I ask because i saw in the more of the books that the substitutions are generally like this $y=ux$ or $y=u/x$, so i am not sure.

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    $\begingroup$ Nonlinear in $x$ is OK if it works! Even $y = u/x$ is nonlinear in $x$. $\endgroup$ – Robert Lewis Oct 2 '17 at 3:32
  • $\begingroup$ ohh, I wanted to write $y=x/u$, but thanks, is good to know that that substitution works. $\endgroup$ – Gabriel Sandoval Oct 2 '17 at 3:36
  • $\begingroup$ Not only it works but this is a very clever substitution ! $\to +1$ $\endgroup$ – Claude Leibovici Oct 2 '17 at 4:24
  • $\begingroup$ Thanks so much, bro. $\endgroup$ – Gabriel Sandoval Oct 2 '17 at 5:03

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