Rewrite: \(\displaystyle \L\ \frac{dy}{dx}\ + 4xy = (\frac{cosx}{e^{2x^2}}\)\)
Take your integrating factor, \(\displaystyle \L\ e^{2x^2}\)
Hence, \(\displaystyle \L\ \frac{d}{dx}\ (y e^{2x^2}) = cosx\)
And, integrating: \(\displaystyle \L\ y e^{2x^2} = sinx + C\)
Therefore: \(\displaystyle \L\ y = sinx e^{-2x^2} + Ce^{-2x^2}\)
\(\displaystyle \L\ = e^{-2x^2}(sinx + C)\)