Answer
$$ y^2-2y= t^2+14.$$
Work Step by Step
By separation of variables, we have
$$(y-1)dy=t dt $$
then by integration, we get
$$ \frac{1}{2}y^2-y=\frac{1}{2}t^2+c \Longrightarrow y^2-2y= t^2+A .$$
Now, since $y(1)=-3$, then $A=14$.
So we have $$ y^2-2y= t^2+14.$$