Question $[30$ points$]$

$(a)$ Solve the linear differential equation

$x\frac{dy}{dx}+3y=x-3$

$(b)$ Given the initial condition $y(1)=3to$ the

differential equation $in(a),$ find the explicit solution $of$

the initial$-$value problem. What $is$ the largest interval

over which the solution $is$ defined?