Differential Equations | Part IA, 2006

Obtain a power series solution of the problem

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

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

Let y0(x),y1(x),y2(x),y_{0}(x), y_{1}(x), y_{2}(x), \ldots be defined recursively as follows: y0(x)=xy_{0}(x)=x. Given yn1(x)y_{n-1}(x), define yn(x)y_{n}(x) to be the solution of

xyn(x)=yn1,yn(0)=0,yn(0)=1x y_{n}^{\prime \prime}(x)=-y_{n-1}, \quad y_{n}(0)=0, y_{n}^{\prime}(0)=1

By calculating y1,y2,y3y_{1}, y_{2}, y_{3}, or otherwise, obtain and prove a general formula for yn(x)y_{n}(x). Comment on the relation to the power series solution obtained previously.

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