Chapter 11 - Series Solutions to Linear Differential Equations - 11.2 Series Solutions about an Ordinary Point - Problems - Page 740: 5

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Given: $y''+xy=0$ Let $y=\sum a_nx^n$, obtain: $y''+xy$$=\sum k(k-1)a_{k}x^{k-2}+x\sum a_{k}x^{k}\\ =2a_2+\sum [(k+2)(k+3)a_{k+3}+a_k]x^{k+1}\\ =0$ Comparing $x^k$ on both sides: $a_2=0\\ a_{k+3}=-\frac{1}{(k+2)(k+3}a_k$ It gives: $a_3=-\frac{1}{6}a_0\\ a_6=\frac{1}{180}a_0\\ a_9=-\frac{1}{12960}a_0\\ a_{12}=\frac{1}{1710720}a_0$ and: $a_4=a_1\\ a_7=-\frac{1}{12}a_1\\ a_{10}=\frac{1}{45360}a_1$ Therefore: $f(x)=a_0 (1-\frac{x^3}{6}+\frac{x^6}{180}-\frac{x^9}{12960}+\frac{x^{12}}{1710720}...)+a_1(x-\frac{x^4}{12}+\frac{x^7}{504}-\frac{x^{10}}{45360}...)$