2.II.8B

Obtain a power series solution of the problem

$x y^{\prime \prime}+y=0, \quad y(0)=0, y^{\prime}(0)=1$

[You need not find the general power series solution.]

Let $y_{0}(x), y_{1}(x), y_{2}(x), \ldots$ be defined recursively as follows: $y_{0}(x)=x$. Given $y_{n-1}(x)$, define $y_{n}(x)$ to be the solution of

$x y_{n}^{\prime \prime}(x)=-y_{n-1}, \quad y_{n}(0)=0, y_{n}^{\prime}(0)=1$

By calculating $y_{1}, y_{2}, y_{3}$, or otherwise, obtain and prove a general formula for $y_{n}(x)$. Comment on the relation to the power series solution obtained previously.

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