4.35.33 \(a x \left (1-x^2\right ) y'(x)+b y(x)+\left (x^2+1\right )^2 y''(x)=0\)

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

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 2.44551 (sec), leaf count = 0 , DifferentialRoot result

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

Maple ✓
cpu = 2.355 (sec), leaf count = 110

\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{\frac {a}{x^{2}+1}} \HeunC \left (a , \frac {a}{2}+\frac {1}{2}, -\frac {1}{2}, a +\frac {1}{4} a^{2}, -\frac {7}{8} a -\frac {1}{4} a^{2}+\frac {1}{8}-\frac {1}{4} b , \frac {1}{x^{2}+1}\right )+\textit {\_C2} \left (x^{2}+1\right )^{\frac {a}{2}+\frac {1}{2}} {\mathrm e}^{\frac {a}{x^{2}+1}} \HeunC \left (a , -\frac {a}{2}-\frac {1}{2}, -\frac {1}{2}, a +\frac {1}{4} a^{2}, -\frac {7}{8} a -\frac {1}{4} a^{2}+\frac {1}{8}-\frac {1}{4} b , \frac {1}{x^{2}+1}\right )\right ]\] Mathematica raw input

DSolve[b*y[x] + a*x*(1 - x^2)*y'[x] + (1 + x^2)^2*y''[x] == 0,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve((x^2+1)^2*diff(diff(y(x),x),x)+a*x*(-x^2+1)*diff(y(x),x)+b*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1*exp(a/(x^2+1))*HeunC(a,1/2*a+1/2,-1/2,a+1/4*a^2,-7/8*a-1/4*a^2+1/8-1
/4*b,1/(x^2+1))+_C2*(x^2+1)^(1/2*a+1/2)*exp(a/(x^2+1))*HeunC(a,-1/2*a-1/2,-1/2,a
+1/4*a^2,-7/8*a-1/4*a^2+1/8-1/4*b,1/(x^2+1))]