\(\(\( (*\ 
      our\ new\ function\ *) \)\(
\)\(\(c\_2[
          x] = \((2 + x\^2)\) a[x];\)
 (*\ 
      cache\ some\ derivatives\ *) 

    \(d1 = D[c\_2[x], x];\)

    \(d2 = D[c\_2[x], {x, 2}];\)\)\)\)

d2 + (1/2) x d1 - (Subscript[c, 2])[x] // Simplify

i1 = 
      Integrate[((-1)/2) (((x (10 + (x^2)))/((2 + (x^2))))), 
        x]

c2  = 
      Integrate[(1/(((2 + (x^2)))^2)) (E^(((-(x^2))/4))), x]

(* 
      our solution c[
          x] up to this point is *) 
c = -1 + A (2 + (x^2)) + B (2 + (x^2)) c2 // Simplify

c /. {x -> 0}

c = 
      1 + (x^2) + B (2 + (x^2)) c2 // Simplify

bSol = Solve[c == 0, {B}]

(* 
      take the derivative to get one boundary conditon *) 
dbc = D[c, x] // Simplify

(dbc /. bSol)[[1]] // Simplify