4.44.22 \(\left (x^2+1\right ) y'''(x)+8 x y'(x)+10 y'(x)=0\)

ODE
\[ \left (x^2+1\right ) y'''(x)+8 x y'(x)+10 y'(x)=0 \] ODE Classification

[[_3rd_order, _missing_y]]

Book solution method
TO DO

Mathematica ✗
cpu = 0.751424 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left [\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\left (\unicode {f817}^2+1\right ) \unicode {f818}^{\text {Symbol}[\text {StringJoin}[\text {ConstantArray}[\prime ,3]]]}(\unicode {f817})+(8 \unicode {f817}+10) \unicode {f818}'(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2,\unicode {f818}''(0)=c_3\right \}\right ][x]\right \}\right \}\]

Maple ✓
cpu = 0.162 (sec), leaf count = 80

\[ \left \{ y \left ( x \right ) ={\it \_C1}+{\it \_C2}\,\int \! \left ( x+i \right ) {\it HeunC} \left ( 0,1,-1,16\,i,{\frac {21}{2}}-8\,i,{\frac {1}{2}}-{\frac {i}{2}}x \right ) \,{\rm d}x+{\it \_C3}\,\int \! \left ( x+i \right ) {\it HeunC} \left ( 0,1,-1,16\,i,{\frac {21}{2}}-8\,i,{\frac {1}{2}}-{\frac {i}{2}}x \right ) \int \!{\frac {1}{ \left ( x+i \right ) ^{2}} \left ( {\it HeunC} \left ( 0,1,-1,16\,i,{\frac {21}{2}}-8\,i,{\frac {1}{2}}-{\frac {i}{2}}x \right ) \right ) ^{-2}}\,{\rm d}x\,{\rm d}x \right \} \] Mathematica raw input

DSolve[10*y'[x] + 8*x*y'[x] + (1 + x^2)*y'''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(10 + 8*\[FormalX
])*Derivative[1][\[FormalY]][\[FormalX]] + (1 + \[FormalX]^2)*Derivative[3][\[Fo
rmalY]][\[FormalX]] == 0, \[FormalY][0] == C[1], Derivative[1][\[FormalY]][0] ==
 C[2], Derivative[2][\[FormalY]][0] == C[3]}]][x]}}

Maple raw input

dsolve((x^2+1)*diff(diff(diff(y(x),x),x),x)+8*x*diff(y(x),x)+10*diff(y(x),x) = 0, y(x),'implicit')

Maple raw output

y(x) = _C1+_C2*Int((x+I)*HeunC(0,1,-1,16*I,21/2-8*I,1/2-1/2*I*x),x)+_C3*Int((x+I
)*HeunC(0,1,-1,16*I,21/2-8*I,1/2-1/2*I*x)*Int(1/(x+I)^2/HeunC(0,1,-1,16*I,21/2-8
*I,1/2-1/2*I*x)^2,x),x)