Question

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

Answer 1

Given: $\frac{dy}{dx}+2xy=y$
$\Rightarrow \frac{dy}{dx}=y-2xy$
$\frac{dy}{dx}=y(1-2x)$
$\Rightarrow \frac{dy}{y}=(1-2x)dx$
Integrating on both sides, we get
$\Rightarrow \int \frac{dy}{y}=\int (1-2x)dx$
$\Rightarrow \ln |y|=x-2\cdot \frac{x^2}{2}+c$
$\Rightarrow \ln |y|=x-x^2+c$
$\Rightarrow |y|=e^{x-x^2+c}=e^{x-x^2}\left(e^c\right)$
$\Rightarrow y=\left(e^c\right)e^{x-x^2}=k{e}^{x-x^2}$
Hence, the required solution is
$y=k{e}^{(x-x^2)}$