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 = 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

\[ \left \{ y \left ( x \right ) ={ \left ( 1+ix \right ) ^{{\frac {3}{4}}+{\frac {i}{4}}} \left ( \left ( x+i \right ) ^{{\frac {1}{2}}-{\frac {i}{4}}}{\it HeunG} \left ( 2,1+i,1,1,{\frac {3}{2}}-{\frac {i}{2}},0,1-ix \right ) {\it \_C1}+ \left ( x+i \right ) ^{{\frac {i}{4}}}{\it HeunG} \left ( 2,{\frac {3\,i}{2}},{\frac {1}{2}}+{\frac {i}{2}},{\frac {1}{2}}+{\frac {i}{2}},{\frac {1}{2}}+{\frac {i}{2}},0,1-ix \right ) {\it \_C2} \right ) {{\rm e}^{-{\frac {\arctan \left ( x \right ) }{2}}}}{\frac {1}{\sqrt [4]{x-i}}}} \right \} \] 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*(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)