$(25$ points$)$ Solve the following differential equation using the method

$of$ separable variables.

$\frac{dy}{dt}=\frac{t-2}{y-1}$

Then discuss the existence and uniqueness $of$ the solutions for the following

initial conditions:

$A.y(2)=0,$

$B.y(2)=1,$

$C.y(0)=1.$