Question

The general solution of the differential equation $\frac{dy}{dx}+2xy=y$ is (where $c$ is a constant of integration)

• A. $|y|=c{e}^{x-x^2}$
• B. $y=c{e}^{x^2-x}$
• C. $y=c{e}^{|x|}$
• D. $|y|=c{e}^{-x^2}$

Answer 1

The correct option is A $|y|=c{e}^{x-x^2}$

$\frac{dy}{dx}+2xy=y$
$\Rightarrow \frac{dy}{dx}=y(1-2x)$
$\Rightarrow \frac{dy}{y}=(1-2x)dx$
on integrating it, we get
$\Rightarrow \ln |y|=x-x^2+c_1$
$\Rightarrow |y|=e^{x-x^2}{e}^{c_1}=c{e}^{x-x^2}$ where $c=e^{c_1}$
$\Rightarrow |y|=c{e}^{x-x^2}$ is the required solution.