\[
\begin{array}{l}
y^{\prime \prime}-2 x y^{\prime}+y=0 \\
y=\sum^{\infty} a_{n} x^{n} \quad y^{\prime}=\sum^{\infty} n x_{n}^{n-1} \quad y^{\prime \prime}=\sum^{\infty} n=x^{n-2} \quad K=n-2 \rightarrow n=k+2 \\
y=\sum_{n=0}^{\infty} a_{n} x^{n}, y^{\prime}=\sum_{n=1}^{\infty} n a_{n} x^{n-1}, y^{\prime \prime}=\sum_{n=2}^{\infty} n(n-1) a_{n} x^{n-2} \\
k=0, n=2 \\
\sum_{n=2}^{\infty} n(n-1) a_{n} x^{n-2}-\sum_{n=1}^{\infty} 2 n a_{n} x^{n}+\sum_{n=0}^{\infty} a_{n} x^{n}=0 \\
k=n-1 \rightarrow n=k+1 \\
\sum_{k=0}^{\infty}(k+2)(k+1) a_{k+2} x^{k}-\sum_{k=0}^{\infty} 2(k+1) a_{k+1} x^{k}+\sum_{k=0}^{\infty} a_{k} x^{k}=0 \\
\sum_{n=0}^{\infty}(n+2)(n+1) a_{n+2} x^{n}-\sum_{n=0}^{\infty} 2(n+1) a_{n+1} x^{n}+\sum_{n=0}^{\infty} a_{n} x^{n}=0 \\
\sum_{n=0}^{\infty}\left[(n+2)(n+1) a_{n+2}-2(n+1) a_{n+1}+a_{n}\right] x^{n}=0 \\
(n+2)(n+1) a_{n+2}-2(n+1) a_{n+1}+a_{n}=0 \\
a_{n+2}=\frac{2(n+1) a_{n+1}-a_{n}}{(n+2)(n+1)} \\
n=0 \rightarrow a_{2}=\frac{2 a_{1}-a_{0}}{4 a_{2}^{2}-a}=a_{1}-\frac{a_{0}}{2} \\
n=1 \longrightarrow a_{3}=\frac{4 a_{2}^{2}-a_{1}}{6 a_{1}-a_{1}}=\frac{2 a_{3}-a_{1}^{2}}{3} \\
n=5 \rightarrow a_{7}=\frac{12 a_{6}-a_{5}}{7.6}=\frac{2 a_{6}-a_{5}}{7} \\
n=2 \longrightarrow a_{4}=\frac{6 a_{3}-a_{2}}{12}=\frac{a_{3}-a_{2}}{2} \\
n=6 \longrightarrow a_{B}=\frac{14 a_{2}-a_{0}}{8-7}=\frac{a_{1}-a_{6}}{4} \\
n=3 \rightarrow a_{5}=\frac{8 a_{4}-2}{5-4}=\frac{2 a_{4}^{2}-a_{3}}{5} \\
n=7 \longrightarrow a_{a}=\frac{16 a_{6}-a_{7}}{96}=\frac{2 a_{8}-a_{7}}{9} \\
n=4 \longrightarrow a_{6}=\frac{10 a_{8}^{5 .-a 4}}{65}=\frac{a_{9}^{5} \cdot a_{4}}{3} \\
n=8 \longrightarrow a_{10}=\frac{18 a_{4}-a_{8}}{10.4}=\frac{a_{4}-a_{8}}{5} \\
n=9 \longrightarrow a_{11}=\frac{20 a_{0}-a_{n}}{11 \cdot 10}=\frac{2 a_{0} \cdot a_{n}}{11}
\end{array}
\]
