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. $x + y = cx^2.$

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