ODE
\[ y(x) \left (\text {a0}+4 \text {a1} \sin ^2(x)-\text {a2} \csc ^2(x)\right )+y''(x)+\cot (x) y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.888692 (sec), leaf count = 191
\[\left \{\left \{y(x)\to e^{2 i \sqrt {\text {a1}} \cos (x)} \left (-\cot ^2\left (\frac {x}{2}\right )\right )^{\frac {\sqrt {\text {a2}}}{2}} \left (c_1 \text {HeunC}\left [\text {a0}+4 i \sqrt {\text {a1}} \left (\sqrt {\text {a2}}-1\right ),-8 i \sqrt {\text {a1}},1-\sqrt {\text {a2}},\sqrt {\text {a2}}+1,-8 i \sqrt {\text {a1}},\sin ^2\left (\frac {x}{2}\right )\right ]+c_2 (\cos (x)-1)^{\sqrt {\text {a2}}} \text {HeunC}\left [\text {a0}-\left (\sqrt {\text {a2}}+1\right ) \left (\sqrt {\text {a2}}+4 i \sqrt {\text {a1}}\right ),-8 i \sqrt {\text {a1}} \left (\sqrt {\text {a2}}+1\right ),\sqrt {\text {a2}}+1,\sqrt {\text {a2}}+1,-8 i \sqrt {\text {a1}},\sin ^2\left (\frac {x}{2}\right )\right ]\right )\right \}\right \}\]
Maple ✓
cpu = 2.546 (sec), leaf count = 67
\[\left [y \left (x \right ) = \textit {\_C1} \left (\sin ^{\sqrt {\mathit {a2}}}\left (x \right )\right ) \HeunC \left (0, \sqrt {\mathit {a2}}, -\frac {1}{2}, -\mathit {a1} , \frac {\mathit {a2}}{4}+\frac {1}{4}-\frac {\mathit {a0}}{4}, \sin ^{2}\left (x \right )\right )+\textit {\_C2} \left (\sin ^{-\sqrt {\mathit {a2}}}\left (x \right )\right ) \HeunC \left (0, -\sqrt {\mathit {a2}}, -\frac {1}{2}, -\mathit {a1} , \frac {\mathit {a2}}{4}+\frac {1}{4}-\frac {\mathit {a0}}{4}, \sin ^{2}\left (x \right )\right )\right ]\] Mathematica raw input
DSolve[(a0 - a2*Csc[x]^2 + 4*a1*Sin[x]^2)*y[x] + Cot[x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> E^((2*I)*Sqrt[a1]*Cos[x])*(-Cot[x/2]^2)^(Sqrt[a2]/2)*(C[1]*HeunC[a0 + (4*I)*Sqrt[a1]*(-1 + Sqrt[a2]), (-8*I)*Sqrt[a1], 1 - Sqrt[a2], 1 + Sqrt[a2], (-8*I)*Sqrt[a1], Sin[x/2]^2] + C[2]*(-1 + Cos[x])^Sqrt[a2]*HeunC[a0 - (1 + Sqrt[a2])*((4*I)*Sqrt[a1] + Sqrt[a2]), (-8*I)*Sqrt[a1]*(1 + Sqrt[a2]), 1 + Sqrt[a2], 1 + Sqrt[a2], (-8*I)*Sqrt[a1], Sin[x/2]^2])}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+cot(x)*diff(y(x),x)+(a0+4*a1*sin(x)^2-a2*csc(x)^2)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*sin(x)^(a2^(1/2))*HeunC(0,a2^(1/2),-1/2,-a1,1/4*a2+1/4-1/4*a0,sin(x)^2)+_C2*sin(x)^(-a2^(1/2))*HeunC(0,-a2^(1/2),-1/2,-a1,1/4*a2+1/4-1/4*a0,sin(x)^2)]