4.34.45 \(x \left (x^2+1\right ) y''(x)+x (x+1) y'(x)+y(x)=0\)

ODE
\[ x \left (x^2+1\right ) y''(x)+x (x+1) y'(x)+y(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 0.846006 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to (x)\right \}\right \}\]

Maple ✓
cpu = 1.642 (sec), leaf count = 103

\[\left [y \left (x \right ) = \frac {\textit {\_C1} \left (i x +1\right )^{\frac {3}{4}+\frac {i}{4}} \left (x +i\right )^{\frac {1}{2}-\frac {i}{4}} {\mathrm e}^{-\frac {\arctan \left (x \right )}{2}} \mathit {HG}\left (2, 1+i, 1, 1, \frac {3}{2}-\frac {i}{2}, 0, -i x +1\right )}{\left (x -i\right )^{\frac {1}{4}}}+\frac {\textit {\_C2} \left (i x +1\right )^{\frac {3}{4}+\frac {i}{4}} \left (x +i\right )^{\frac {i}{4}} {\mathrm e}^{-\frac {\arctan \left (x \right )}{2}} \mathit {HG}\left (2, \frac {3 i}{2}, \frac {1}{2}+\frac {i}{2}, \frac {1}{2}+\frac {i}{2}, \frac {1}{2}+\frac {i}{2}, 0, -i x +1\right )}{\left (x -i\right )^{\frac {1}{4}}}\right ]\] 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*(x+1)*diff(y(x),x)+y(x) = 0, y(x))

Maple raw output

[y(x) = _C1/(x-I)^(1/4)*(1+I*x)^(3/4+1/4*I)*(x+I)^(1/2-1/4*I)*exp(-1/2*arctan(x)
)*HeunG(2,1+I,1,1,3/2-1/2*I,0,1-I*x)+_C2/(x-I)^(1/4)*(1+I*x)^(3/4+1/4*I)*(x+I)^(
1/4*I)*exp(-1/2*arctan(x))*HeunG(2,3/2*I,1/2+1/2*I,1/2+1/2*I,1/2+1/2*I,0,1-I*x)]