Solve the given initial value $DE$ using the linear first$-$order method $(also$ known $as$ "integrating factor method"$).$

$x(x+1)\frac{dy}{dx}+xy=1,y(e)=1$

Possible Answers

$(a.)y=\frac{lnx-e}{x+1}$

$(b.)y=\frac{lnx+e}{x+1}$

$(c.)y=\frac{lnx+e}{x-1}$

$(d.)y=\frac{lnx-e}{x-1}$

$(e.)$ None $of$ the above