1
$\begingroup$

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.

  • 0
    We tend to avoid titles consist of only latex code due to some browser rendering difficulties.2011-12-01

2 Answers 2

2

HINT

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

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

1

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 $$ y=x\left(\frac15x^{5/2}+c\right)^2. $$