4.34.45 x(x2+1)y(x)+x(x+1)y(x)+y(x)=0

ODE
x(x2+1)y(x)+x(x+1)y(x)+y(x)=0 ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 1.36459 (sec), leaf count = 0 , DifferentialRoot result

\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\unicode {f818}(\unicode {f817})+\left (\unicode {f817}^2+\unicode {f817}\right ) \unicode {f818}'(\unicode {f817})+\left (\unicode {f817}^3+\unicode {f817}\right ) \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(1)=c_1,\unicode {f818}'(1)=c_2\right \}\right )(x)\right \}\right \}

Maple
cpu = 0.361 (sec), leaf count = 83

{y(x)=(1+ix)34+i4((x+i)12i4HeunG(2,1+i,1,1,32i2,0,1ix)_C1+(x+i)i4HeunG(2,3i2,12+i2,12+i2,12+i2,0,1ix)_C2)earctan(x)21xi4} Mathematica raw input

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)