Question

Solve the differential equation dy/dx+4xy=x

Answer 1

$\dfrac{\mathrm dy}{\mathrm dx}+4xy=x$
$\dfrac{\mathrm dy}{\mathrm dx}=x(1-4y)$

This ODE is separable, so you can write

$\dfrac{\mathrm dy}{1-4y}=x\,\mathrm dx$

Integrating both sides gives

$-\dfrac14\ln|1-4y|=\dfrac12x^2+C$
$\ln|1-4y|=-2x^2+C$
$1-4y=e^{-2x^2+C}$
$1-4y=Ce^{-2x^2}$
$y=\dfrac{1-Ce^{-2x^2}}4$