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Which technique should I use for solving the follwoing DE?

$$ y' - \frac{1}{x} y = x^2\sqrt{y} $$ I have tried some algebraic manipulations but I could not recognize any pattern.

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We tend to avoid titles consist of only latex code due to some browser rendering difficulties. –  user13838 Dec 1 '11 at 19:33
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2 Answers

up vote 2 down vote accepted


  1. Divide by $\sqrt{y}$.

  2. Think of the chain rule and make a substitution...

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First, $z=y/x$ yields $z'=x\sqrt{y}=x^{3/2}\sqrt{z}$. Then $u=\sqrt{z}$ yields $u'=\frac12x^{3/2}$ hence $u=\frac15x^{5/2}+c$. Finally, $y=xz=xu^2$ hence $$ y=x\left(\frac15x^{5/2}+c\right)^2. $$

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