Take the 2-minute tour ×  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.  share|improve this question We tend to
avoid titles consist of only latex code due to some browser
rendering difficulties. –  user13838 Dec 1 '11 at 19:33 add
comment  2 Answers  up vote 2 down vote accepted     1. Divide
by $\sqrt{y}$.    2. Think of the chain rule and make a
substitution...  share|improve this answer add comment  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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