4.35.35 \(x^2 \left (a^2+x^2\right )^2 y''(x)+x \left (a^2+2 x^2\right ) y'(x)+b^2 y(x)=0\)

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

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 10.4984 (sec), leaf count = 0 , DifferentialRoot result

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

Maple ✓
cpu = 2.791 (sec), leaf count = 295

\[\left [y \left (x \right ) = \textit {\_C1} \,x^{\frac {a^{2}+\sqrt {a^{4}-2 a^{2}-4 b^{2}+1}-1}{2 a^{2}}} \left (a^{2}+x^{2}\right )^{-\frac {-a^{2}+\sqrt {a^{4}-2 a^{2}-4 b^{2}+1}-1}{4 a^{2}}} {\mathrm e}^{\frac {1}{2 a^{2}+2 x^{2}}} \HeunC \left (\frac {1}{2 a^{2}}, -\frac {1}{2}, \frac {\sqrt {a^{4}-2 a^{2}-4 b^{2}+1}}{2 a^{2}}, \frac {4 a^{2}+1}{8 a^{4}}, \frac {a^{2}-6}{8 a^{2}}, \frac {a^{2}}{a^{2}+x^{2}}\right )+\textit {\_C2} \,x^{\frac {a^{2}+\sqrt {a^{4}-2 a^{2}-4 b^{2}+1}-1}{2 a^{2}}} \left (a^{2}+x^{2}\right )^{-\frac {a^{2}+\sqrt {a^{4}-2 a^{2}-4 b^{2}+1}-1}{4 a^{2}}} {\mathrm e}^{\frac {1}{2 a^{2}+2 x^{2}}} \HeunC \left (\frac {1}{2 a^{2}}, \frac {1}{2}, \frac {\sqrt {a^{4}-2 a^{2}-4 b^{2}+1}}{2 a^{2}}, \frac {4 a^{2}+1}{8 a^{4}}, \frac {a^{2}-6}{8 a^{2}}, \frac {a^{2}}{a^{2}+x^{2}}\right )\right ]\] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = _C1*x^(1/2*(a^2+(a^4-2*a^2-4*b^2+1)^(1/2)-1)/a^2)*(a^2+x^2)^(-1/4*(-a^2+
(a^4-2*a^2-4*b^2+1)^(1/2)-1)/a^2)*exp(1/(2*a^2+2*x^2))*HeunC(1/2/a^2,-1/2,1/2*(a
^4-2*a^2-4*b^2+1)^(1/2)/a^2,1/8/a^4*(4*a^2+1),1/8/a^2*(a^2-6),a^2/(a^2+x^2))+_C2
*x^(1/2*(a^2+(a^4-2*a^2-4*b^2+1)^(1/2)-1)/a^2)*(a^2+x^2)^(-1/4*(a^2+(a^4-2*a^2-4
*b^2+1)^(1/2)-1)/a^2)*exp(1/(2*a^2+2*x^2))*HeunC(1/2/a^2,1/2,1/2*(a^4-2*a^2-4*b^
2+1)^(1/2)/a^2,1/8/a^4*(4*a^2+1),1/8/a^2*(a^2-6),a^2/(a^2+x^2))]