For the linear differential equation $y^\prime+3 xy = x^6e^{\frac{-3x^{2}}{2}}$ the integrating factor is:
After multiplying both sides by the integrating factor and unapplying the product rule we get the new differential equation:
$\frac{d}{dx}\big[$ $\big]$ =
Integrating both sides we get the algebraic equation =
Solving for y, the solution to the differential equation is y = (using k as the constant)