x2y'+xy=x+1; divide through by x:
xy'+y=(x+1)/x=1+1/x,
(d/dx)(xy)=1+1/x,
xy=∫(1+1/x)dx=x+ln|x|+C, where C is the constant of integration.
y=1+(ln|x|+C)/x which can be written y=1+ln|ax|/x where a is a constant (a=eC).
Note that x cannot be zero, so the y-axis is the vertical asymptote. This is a discontinuity. y=1 is the horizontal asymptote.
The largest interval I=(-∞,0) or (0,∞) which have the same size. The transient term is ln|ax|/x which approaches zero as x→∞, and the graph is levelling out to y=1.