4.27.17 \(a \cot (x) y'(x)+y(x) \left (b+k^2 \cos ^2(x)\right )+y''(x)=0\)

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

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 7.3687 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to c_2 \left (e^{i x}\right )+c_1 \left (e^{i x}\right )\right \}\right \}\]

Maple ✓
cpu = 2.237 (sec), leaf count = 77

\[\left [y \left (x \right ) = \textit {\_C1} \HeunC \left (0, \frac {a}{2}-\frac {1}{2}, -\frac {1}{2}, \frac {k^{2}}{4}, -\frac {a}{8}+\frac {3}{8}-\frac {k^{2}}{4}-\frac {b}{4}, \sin ^{2}\left (x \right )\right )+\textit {\_C2} \HeunC \left (0, -\frac {a}{2}+\frac {1}{2}, -\frac {1}{2}, \frac {k^{2}}{4}, -\frac {a}{8}+\frac {3}{8}-\frac {k^{2}}{4}-\frac {b}{4}, \sin ^{2}\left (x \right )\right ) \left (\sin ^{1-a}\left (x \right )\right )\right ]\] Mathematica raw input

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

Mathematica raw output

{{y[x] -> C[2]*DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {-((-1 + \[Fo
rmalX])*(1 + \[FormalX])*(4*\[FormalX]^2*b + k^2 + 2*\[FormalX]^2*k^2 + \[Formal
X]^4*k^2)*\[FormalY][\[FormalX]]) + 4*\[FormalX]^3*(-1 + \[FormalX]^2 + a + \[Fo
rmalX]^2*a)*Derivative[1][\[FormalY]][\[FormalX]] + (-4*\[FormalX]^4 + 4*\[Forma
lX]^6)*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][2] == 0, Derivativ
e[1][\[FormalY]][2] == 1}]][E^(I*x)] + C[1]*DifferentialRoot[Function[{\[FormalY
], \[FormalX]}, {-((-1 + \[FormalX])*(1 + \[FormalX])*(4*\[FormalX]^2*b + k^2 + 
2*\[FormalX]^2*k^2 + \[FormalX]^4*k^2)*\[FormalY][\[FormalX]]) + 4*\[FormalX]^3*
(-1 + \[FormalX]^2 + a + \[FormalX]^2*a)*Derivative[1][\[FormalY]][\[FormalX]] +
 (-4*\[FormalX]^4 + 4*\[FormalX]^6)*Derivative[2][\[FormalY]][\[FormalX]] == 0, 
\[FormalY][2] == 1, Derivative[1][\[FormalY]][2] == 0}]][E^(I*x)]}}

Maple raw input

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

Maple raw output

[y(x) = _C1*HeunC(0,1/2*a-1/2,-1/2,1/4*k^2,-1/8*a+3/8-1/4*k^2-1/4*b,sin(x)^2)+_C
2*HeunC(0,-1/2*a+1/2,-1/2,1/4*k^2,-1/8*a+3/8-1/4*k^2-1/4*b,sin(x)^2)*sin(x)^(1-a
)]