Question

Solve:$\frac{dy}{dx}=\frac{y}{x-\sqrt{xy}}$

Answer 1

$\frac{dy}{dx}=\frac{y}{x-\sqrt{xy}}Let,y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}hence,v+x\frac{dv}{dx}=\frac{vx}{x-\sqrt{x^{2}v}}\Rightarrow \left(\frac{1-\sqrt{v}}{v^{\frac{3}{2}}}\right)dv=\frac{dx}{x}\Rightarrow (\frac{1}{v^{\frac{3}{2}}}-\frac{1}{v})dv=\frac{dx}{x}\Rightarrow \int (\frac{1}{v^{\frac{3}{2}}}-\frac{1}{v})dv=\int \frac{dx}{x}\Rightarrow \frac{2}{\sqrt{v}}-\log v=\log x+C\Rightarrow \frac{2}{\sqrt{\frac{y}{x}}}-\log \left(\frac{y}{x}\right)=\log x+C\Rightarrow 2\sqrt{\frac{x}{y}}=\log \left(\frac{y}{x}\right)+\log x+C\Rightarrow 2\sqrt{\frac{x}{y}}=\log y+C$