Question

The general solution of the differential equation $\frac{dy}{dx}=\frac{x^2+xy+y^2}{x^2}$ is
(where $c$ is constant of integration)

• A. ${\tan }^{-1}\left(\frac{x}{y}\right)=\ln |x|+c$
• B. ${\tan }^{-1}\left(\frac{y}{x}\right)=\ln |x|+c$
• C. ${\tan }^{-1}\left(\frac{y}{x}\right)=\ln |y|+c$
• D. ${\tan }^{-1}\left(\frac{x}{y}\right)=\ln |y|+c$

Answer 1

The correct option is B ${\tan }^{-1}\left(\frac{y}{x}\right)=\ln |x|+c$

Let $y=vx$
$\frac{dy}{dx}=v+x\frac{dv}{dx}$
Given DE can be convertable as
$v+x\frac{dv}{dx}=1+v+v^2$
$\Rightarrow \frac{dv}{1+v^2}=\frac{dx}{x}$
$\text{On integrating both sides, we have}$
${\tan }^{-1}v=\ln |x|+c$
$\Rightarrow {\tan }^{-1}\left(\frac{y}{x}\right)=\ln |x|+c$