Question

General solution of the differential equation $\frac{dy}{dx}=1+xy$ is

• A. $y=c\cdot e^{-x2/2}$
• B. $y=c\cdot e^{x2/2}$
• C. $y=(x+c)\cdot e^{-x2/2}$
• D. None

Answer 1

The correct option is C $y=(x+c)\cdot e^{-x2/2}$

$\frac{dy}{dx}=1+xy\Rightarrow \frac{dy}{dx}-ky=1$

$IF=e^{\int -xdx}=e^{-\frac{x^2}{2}}$

$\frac{dy}{dx}=1+xy\Rightarrow e^{-\frac{x^2}{2}}\frac{dy}{dx}-2yx{e}^{-\frac{x^2}{2}}=1.e^{-\frac{x^2}{2}}$

$\therefore \frac{d}{dx}\left(y{e}^{-\frac{x^2}{2}}\right)=e^{-\frac{x^2}{2}}$

$y{e}^{-\frac{x^2}{2}}=e^{-\frac{x^2}{2}}+c$