$$ $\backslash$begin$\{$array$\}\{111)$ Thline $\backslash$text $\{$ Find the general solution of $\}$ y$^\{\backslash$prime $\backslash$prime$)-$x$^\{2\}$ y$^\{\backslash$prime$)-$y$=0\backslash$text $\{$ by power seri$)$ Whline $\backslash$text $\{$A$\}$ & $\backslash$text $\{$ RF $($recurrence formula$)$: $\}$ a$\_\{$n$+2\}=\backslash$frac$\{-1\}($n$+2)($n$+1)\}$ a$\_\{$n$-1\}$ &y$=$a$\_\{0\}\backslash$left$(1-\backslash$frac$\{1\}\{6\}$ x$^\{3\}+\backslash$frac$\{1\}\{180)**\{6\}+\backslash$cdots$\backslash$right$)+$a$\_\{1\}\backslash$left$($x$-\backslash$frac$\{1\}\{12\}$ x$^\{4\}+\backslash$frac$\{1\}\{504)$ x$^\{7\}+\backslash$cdots$\backslash$right$)$ W $\backslash$text $\{$ B$)\}$ & $\backslash$text $\{$ RF: $\}$ a$\_\{$n$+2)=\backslash$frac$\{2\}\{($n$+2)\}$ a$\_\{$n$\}\backslash$text $\{\}$ & y$=$a$\_\{0\}\backslash$left$(1+$x$^\{2\}+\backslash$frac$\{1\}\{2\}$ x$^\{4\}+\backslash$frac$\{1\}(6)**\{6\}+\backslash$cdots$\backslash$right$)+$a$\_\{1\}\backslash$left$($x$+\backslash$frac$\{2\}\{3\}**\{3\}+\backslash$frac$\{4\}\{15\}**\{5\}+\backslash$frac$\{8\}\{105)**\{7\}+\backslash$cdots$\backslash$right$)$ W & $\backslash$text $\{$ RF: $\}$ a$\_\{$n$+2\}=\backslash$frac$\{-1\}\{($n$+2)\}$ a$\_\{$n$-1\}$ W & y$=$a$\_\{0\}\backslash$left$(1-\backslash$frac$\{1\}\{3\}$ x$^\{3\}+\backslash$frac$\{1\}18)**\{6\}+\backslash$cdots$\backslash$right$)+$a$\_\{1\}\backslash$left$(\backslash$frac$\{1\}\{4\}$ x$^\{4\}+\backslash$frac$\{1\}\{28)$ X$^\{7\}+\backslash$cdots$\backslash$right$)\backslash$text $\{$D $\}$ &$\backslash$text $\{$ RF: $\}$ a$\_\{$n$+2\}=\backslash$frac$\{$n$-1\}\{($n$+2)($n$+1)\}$ a$\_\{$n$-1\}+\backslash$frac$\{1\}($n$+2)($n$+1)\}$ a$\_\{$n$\}$ & $\backslash$quad$,\backslash$quad$,\backslash$end$\{$array $$ SP$.$SD$\mathrm{.}4091$