Find the general solution of the form $\displaystyle

y = \sum_{n = 0}^{\infty}a_nx^n$ for the differential equation $\displaystyle y''+xy'+y=0$ .

Derive from the general solution that e^(-(x^2)/2) is a solution of the differential equation.

I know how to solve the DE using the series, but do not know specifically how to show that one of the solutions is e^(-(x^2)/2) as asked by my teacher. please help in here