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