设幂级数 $y(x)=\sum_{k=0}^{+\infty} a_k x^k(-\infty < x < +\infty)$ 满足微分方程初值问题:
$$
\left\{\begin{array}{l}
y^{\prime \prime}+2 x y^{\prime}+2 y=0 \\
y(0)=1, y^{\prime}(0)=0
\end{array}\right.
$$
(1) 证明: $a_{k+2}=-\frac{2}{k+2} a_k, k=0,1,2, \cdots$;
(2)求 $y(x)$ 的表达式.