Question

The general solution of the differential equation
$\frac{dy}{dx}=e^{\frac{x^2}{2}}+xy$ is

• A. $y=c{e}^{\frac{-x^2}{2}}$
• B. $y=c{e}^{\frac{x^2}{2}}$
• C. $y=\left(x+c\right)e^{\frac{x^2}{2}}$
• D. $y=\left(c-x\right)e^{\frac{x^2}{2}}$

Answer 1

Answer: (C)

$\frac{dy}{dx}=e^{x^2/2}+xy$
$\Rightarrow \frac{dy}{dx}-x\cdot y=e^{x^2/2}$
It is a linear differential equation with
$I.F.=e^{-\int xdx}=e^{-x^2/2}$
Now, solution is
$y\cdot e^{-x^2/2}=\int e^{-x^2/2}{e}^{x^2/2}dx+c$
$\Rightarrow y\cdot e^{-x^2/2}=x+c$
$\Rightarrow y=\left(x+c\right)e^{x^2/2}$