Let F(x)=x + (e^(x-1))/x
Find the largest interval on which F is a solution to the I.V.P
dy/dx + ((1-x)/x)y = 2-x , y(1) = 2
You need to have $\displaystyle x\not =0$. Then you are told that $\displaystyle y(1)=2$ therefore, $\displaystyle 1$ has to be included in that interval.
Thus, this forces for $\displaystyle x>0$. The interval is $\displaystyle (0,\infty)$.