6.16 First-order linear differential equations

It turned out to be possible to solve separable differential equations as we can ‘convert’ them into integrable equations (as explained on slide 6.7). We do something similar for the third class of differential equations we consider, called linear differential equations. Consider the differential equation:

This case looks just as hopeless as the equation on slide 6.7 did. But if we multiply by $e^{x^{2}}$ we obtain $e^{x^{2}}\frac{dy}{dx}+2xe^{x^{2}}y=xe^{x^{2}}$. It now turns out (as if by a miracle) that the left-hand side is $\frac{d}{dx}\left({e^{x^{2}}y}\right)$. Thus we have the equation:

which can be solved by integrating.