[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 1.36459 (sec), leaf count = 0 , DifferentialRoot result
Maple ✓
cpu = 0.361 (sec), leaf count = 83
DSolve[y[x] + x*(1 + x)*y'[x] + x*(1 + x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {\[FormalY][\[Form
alX]] + (\[FormalX] + \[FormalX]^2)*Derivative[1][\[FormalY]][\[FormalX]] + (\[F
ormalX] + \[FormalX]^3)*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][1
] == C[1], Derivative[1][\[FormalY]][1] == C[2]}]][x]}}
Maple raw input
dsolve(x*(x^2+1)*diff(diff(y(x),x),x)+x*(1+x)*diff(y(x),x)+y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (1+I*x)^(3/4+1/4*I)*exp(-1/2*arctan(x))*((x+I)^(1/2-1/4*I)*HeunG(2,1+I,1,
1,3/2-1/2*I,0,1-I*x)*_C1+(x+I)^(1/4*I)*HeunG(2,3/2*I,1/2+1/2*I,1/2+1/2*I,1/2+1/2
*I,0,1-I*x)*_C2)/(x-I)^(1/4)