# What is a solution to the differential equation dy/dy=sqrt(x/y)?

##### 1 Answer
Jul 11, 2016

${y}^{\frac{3}{2}} = {x}^{\frac{3}{2}} + C$

and if you like

$y = {\left({x}^{\frac{3}{2}} + C\right)}^{\frac{2}{3}}$

#### Explanation:

assuming firstly this is a typo and that you meant to say $\frac{\mathrm{dy}}{\textcolor{red}{\mathrm{dx}}}$

if so, separate it

$\frac{\mathrm{dy}}{\mathrm{dx}} = \sqrt{\frac{x}{y}}$

$\sqrt{y} \frac{\mathrm{dy}}{\mathrm{dx}} = \sqrt{x}$

$\int \setminus \sqrt{y} \frac{\mathrm{dy}}{\mathrm{dx}} \setminus \mathrm{dx} = \int \sqrt{x} \setminus \mathrm{dx}$

$\int \setminus \sqrt{y} \setminus \mathrm{dy} = \int \sqrt{x} \setminus \mathrm{dx}$

power rule

$\frac{2}{3} {y}^{\frac{3}{2}} = \frac{2}{3} {x}^{\frac{3}{2}} + C$

${y}^{\frac{3}{2}} = {x}^{\frac{3}{2}} + C$

and if you like

$y = {\left({x}^{\frac{3}{2}} + C\right)}^{\frac{2}{3}}$