Question

Solve:$x^2\frac{dy}{dx}=\frac{y(x+y)}{2}$

Answer 1

$x^2\frac{dy}{dx}=\frac{y(x+y)}{2}let,y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}so,x^2(v+x\frac{dv}{dx})=\frac{vx(x+vx)}{2}\Rightarrow x\frac{dv}{dx}=\frac{v^2-v}{2}\Rightarrow \frac{2dv}{v(v-1)}=\frac{dx}{x}Let,\frac{1}{v(v-1)}=\frac{A}{v}+\frac{B}{v-1}\left(bypartialfraction\right)\Rightarrow 1=(A+B)v-Acomparingbothside,wegetA=-1andB=1\therefore \int \frac{2dv}{v(v-1)}=\int \frac{dx}{x}\Rightarrow \int [\frac{1}{v-1}-\frac{1}{v}]dv=\int \frac{dx}{x}\Rightarrow \log (v-1)-\log v=\log x+\log C\Rightarrow \log \left(\frac{v-1}{v}\right)=\log Cx\Rightarrow \frac{v-1}{v}=Cx\Rightarrow \frac{\frac{y}{x}-1}{\frac{y}{x}}=Cx\Rightarrow y=x+Cxy$