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