Series solutions with Maple
> with(DEtools):
> Order:=10:
> dsolve({diff(y(x),x$2)+x*diff(y(x),x)+y(x)=0,y(2)=3,D(y)(2)=1},y(x),series);

                              2              3               4
y(x) = 3 + x - 2 - 5/2 (x - 2)  + 4/3 (x - 2)  - 1/24 (x - 2)  - 1/4

           5   13         6                7    37         8    71
    (x - 2)  + --- (x - 2)  + 5/504 (x - 2)  - ---- (x - 2)  + -----
               144                             2688            36288

           9            10
    (x - 2)  + O((x - 2)  )

> plot(3+x-2-5/2*(x-2)^2+4/3*(x-2)^3-1/24*(x-2)^4-1/4*(x-2)^5+13/144*(x-2)^6+5/504*(x-2)^7-37/2688*(x-2)^8+71/36288*(x-2)^9,x=0..4,color=black,thickness=3);

We can compare this with a plot of the numerical solution. What happens if x=0..6 instead?
> DEplot({diff(y(x),x$2)+x*diff(y(x),x)+y(x)=0},{y(x)},x=0..4,[[y(2)=3,D(y)(2)=1]]);