$$ \begin{array}{111) Thline \text { Find the general solution of } y^{\prime \prime)-x^{2} y^{\prime)-y=0 \text { by power seri) Whline \text {A} & \text { RF (recurrence formula): } a_{n+2}=\frac{-1}(n+2)(n+1)} a_{n-1} &y=a_{0}\left(1-\frac{1}{6} x^{3}+\frac{1}{180) **{6}+\cdots ight)+a_{1}\left(x-\frac{1}{12} x^{4}+\frac{1}{504) x^{7}+\cdots ight) W \text { B)} & \text { RF: } a_{n+2)=\frac{2} {(n+2)} a_{n} \text { } & y=a_{0}\left(1+x^{2}+\frac{1}{2} x^{4}+\frac{1} (6) **{6}+\cdots ight)+a_{1}\left(x+\frac{2}{3} **{3}+\frac{4}{15} **{5}+\frac{8}{105) **{7}+\cdots ight) W & \text { RF: } a_{n+2}=\frac{-1}{(n+2)} a_{n-1} W & y=a_{0}\left(1-\frac{1}{3} x^{3}+\frac{1}18) **{6}+\cdots ight)+a_{1}\left( \frac{1}{4} x^{4}+\frac{1}{28) X^{7}+\cdots ight) \text {D } &\text { RF: } a_{n+2}=\frac{n-1} {(n+2)(n+1)} a_{n-1}+\frac{1}(n+2)(n+1)} a_{n} & \quad, \quad, \end{array $$ SP.SD.4091