$(1)$ Consider the functions

$y_1=C_1(log(1-t)+log(\frac{1}{1+t}))+C_2$

$y_2=C_1{e}^{-t}cos(\frac{3}{2}{t}^2)+C_2{e}^{-t}sin(\frac{3}{2}{t}^2)$

Use verification by substitution to match each of the above functions with the differential equation which it solves from the following list:

$($a$)$

$($b$)$

$t{y}^{}+(2t-1)y^{}+(9t^3+t-1)y=0$

$(t^2-1)y^{}+2t{y}^{}=0$