Question

Solve $ \frac{dy}{dx}=\frac{{{y}^{2}}}{xy-{{x}^{2}}} $

• A. $ y=c{{e}^{x/y}} $
• B. $ y=c{{e}^{-y/x}}+x $
• C. $ y=c{{e}^{y/x}} $
• D. $ xy=c{{e}^{y/x}} $

Answer 1

Answer: (C)

Given equation is $ \frac{dy}{dx}=\frac{{{y}^{2}}}{xy-{{x}^{2}}} $ It is a homogeneous differential equation
Put $ y=vx $ $ \Rightarrow $ $ \frac{dy}{dx}=v+x\frac{dv}{dx} $
$ \therefore $ $ v+x\frac{dv}{dx}=\frac{{{v}^{2}}{{x}^{2}}}{x\cdot vx-{{x}^{2}}} $
$ \Rightarrow $ $ v+x\frac{dv}{dx}=\frac{{{v}^{2}}}{v-1} $
$ \Rightarrow $ $ x\frac{dv}{dx}=\frac{v}{v-1} $
$ \Rightarrow $ $ \left( 1-\frac{1}{v} \right)dv=\frac{dx}{x} $
On integrating, we get $ v-\log v=\log x-\log c $
$ \Rightarrow $ $ \frac{y}{x}=\log \frac{y}{x}\cdot x\cdot \frac{1}{c} $
$ \Rightarrow $ $ y=c{{e}^{y/x}} $