series solutions

consider

$$y=\sum _{n=0}^{\infty }{a}_n{x}^n$$

then

$$y^{\prime }=\sum _{n=1}^{\infty }n{a}_n{x}^{n-1}=\sum _{n=0}^{\infty }(n+1)a_{n+1}{x}^n$$

and

$$y^{\prime \prime }=\sum _{n=2}^{\infty }n(n-1)a_n{x}^{n-2}=\sum _{n=0}^{\infty }(n+2)(n+1)a_{n+2}{x}^n$$

then solve the equation by normal calculation