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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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2 Answers 2

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HINT

  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. $