$(1)$ Let C$\_(1)$ and C$\_(2)$ be arbitrary constants. The general solution to the homogeneous differential equation $289$x$^(2)$y$^(\text{'}\text{'})+459$xy$^(\text{'})+50$y$=0$ is the function y$($x$)=$C$\_(1)$y$\_(1)($x$)+$C$\_(2)$y$\_(2)($x$)=$C$\_(1)$

$+$C$\_(2)$

$(2)$ The unique solution to the initial value problem

$289$x$^(2)$y$^(\text{'}\text{'})+459$xy$^(\text{'})+50$y$=0,$y$(1)=-4,$y$^(\text{'})(1)=-9$

is the function y$($x$)=$

for xin

For $-\backslash$infty type $-$inf and for $\backslash$infty type inf.