Answer
$$y= 2.$$
Work Step by Step
By separation of variables, we have
$$\frac{dy}{y-2}=(x-1)dx$$
then by integration, we get
$$\ln(y-2)=\frac{1}{2}x^{2}-x+c\Longrightarrow y=Ae^{\frac{1}{2}x^{2}-x}+2.$$
Now, since $y(2)=2$, then $A=0$.
So the general solution is given by $$y= 2.$$
