$(15$ points$)$ Consider

$t^2{x}^{\text{'}\text{'}}+t{x}^{\text{'}}+(t^2-\frac{1}{4})x=0,t>0$

$($a$)$ Use the transformation $x=\frac{y}{\sqrt[\phantom{2}]{t}}$ to write the above equation as an equation of $y.$ This is called the change of variable method. After the transformation, your equation should look like $y^{\text{'}\text{'}}+r(t)y=0$ for some function $r(t).$ Your goal is to find the function $r(t).$

$($b$)$ Find the GS of the transformed equation in the form of $y(t)=c_1{y}_1(t)+c_2{y}_2(t).$

$($c$)$ Use the GS from part $($ii$)$ to find the GS of the original equation.