\(\dfrac{\mathrm dy}{\mathrm dx}+4xy=x\)
\(\dfrac{\mathrm dy}{\mathrm dx}=x(1-4y)\)
This ODE is separable, so you can write
\(\dfrac{\mathrm dy}{1-4y}=x\,\mathrm dx\)
Integrating both sides gives
\(-\dfrac14\ln|1-4y|=\dfrac12x^2+C\)
\(\ln|1-4y|=-2x^2+C\)
\(1-4y=e^{-2x^2+C}\)
\(1-4y=Ce^{-2x^2}\)
\(y=\dfrac{1-Ce^{-2x^2}}4\)
