Question

Solve the differential equation $\frac{dy}{dx}+2xy=y$.

Answer 1

Given that, $\frac{dy}{dx}+2xy=y$
$\Rightarrow \frac{dy}{dx}+2xy-y=0$
$\Rightarrow \frac{dy}{dx}+(2x-1)y=0$
Which is a linear differential equation.
On comparing it with $\frac{dy}{dx}+Py=Q$, we get$P=(2x-1),Q=0$
$IF=e^{\int pdx}=e^{\int (2x-1)dx}$
$=e^{(\frac{2x^2}{2}-x)}$
The complete solution is
$y.e^{x^2-x}=\int Q.e^{x^2-x}=0+C$
$\Rightarrow y.e^{x^2-x}=0+C$
$\Rightarrow y=C{e}^{x-x^2}$