# Solve the differential equation. 5\sqrt{xy}\frac{dy}{dx}=1, x, y>0

Solve the differential equation.
$5\sqrt{xy}\frac{dy}{dx}=1,x,y>0$
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Step 1
Given,
$5\sqrt{xy}\frac{dy}{dx}=1,x,y>0$
Step 2
Rewriting the equation,
$5\sqrt{y}dy=\frac{1}{\sqrt{x}}dx$
Step 3
Integrate both sides,
$⇒5\int \sqrt{y}dy=\int \frac{1}{\sqrt{x}}dx+C$, C is constant.
$⇒5\left(\frac{{y}^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right)=\left(\frac{{x}^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\right)+C$
$⇒5\left(\frac{2{y}^{\frac{3}{2}}}{3}\right)=2{x}^{\frac{1}{2}}+C$
$⇒{y}^{\frac{3}{2}}=\frac{3}{10}\left(2{x}^{\frac{1}{2}}+C\right)$
$⇒y={\left(\frac{3}{10}\left(2{x}^{\frac{1}{2}+C}\right)\right)}^{\frac{2}{3}}$