Question

The solution of the differential equation $\frac{dy}{dx}=\frac{x+2y}{x}$ is

• A. $x+y=c$
• B. $x^2+y^2=c$
• C. $x+y=cx$
• D. $x+y=c{x}^2.$

Answer 1

Answer: (D)

$\frac{dy}{dx}=\frac{x+2y}{x}=1+\frac{2y}{x}$
$\Rightarrow \frac{dy}{dx}-\frac{2}{x}.y=1$
Here $P=-\frac{2}{x}$; $Q=1$
$\therefore \int pdx=\int -\frac{2}{x}dx$
$=-2logx=log{x}^{-2}$
$\therefore e^{\int pdx}=e^{log{x}^{-2}=x^{-2}}$
$\therefore$ Sol. is
$y.\left(x^{-2}\right)=\int x^{-2}dx+c=\frac{x^{-1}}{-1}+c$
$\therefore$ i.e., $\frac{y}{x^2}=-\frac{1}{x}+c$
$\Rightarrow y=-x+c{x}^2$
$\Rightarrow x+y=c{x}^2$.