Solving a 2nd order DE (non-constant coefficients)

Hint:

Notice that $(x^2-1)'=2x$, hence

$$y''+(x^2-1)y'+2xy=(y'+(x^2-1)y)'=0.$$

Par résolution of the first-order linear equation, $$y=e^{x-x^3/3}\left(C_2+C_1\int e^{x^3/3-x}dx\right).$$