4.27.33 \(-a (a+1) y(x) \csc ^2(x)+y''(x)-\tan (x) y'(x)=0\)

ODE
\[ -a (a+1) y(x) \csc ^2(x)+y''(x)-\tan (x) y'(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.503053 (sec), leaf count = 96

\[\left \{\left \{y(x)\to (\cos (x)-1)^{\frac {a+1}{2}} (\cos (x)+1)^{-a/2} \left (c_1 \text {HeunG}\left [2,\frac {1}{4}-a,\frac {1}{2},\frac {3}{2},\frac {1}{2}-a,1,\cos (x)+1\right ]+c_2 (\cos (x)+1)^{a+\frac {1}{2}} \, _2F_1\left (\frac {a+1}{2},\frac {a+2}{2};a+\frac {3}{2};\sin ^2(x)\right )\right )\right \}\right \}\]

Maple ✓
cpu = 1.3 (sec), leaf count = 61

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

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

Mathematica raw output

{{y[x] -> ((-1 + Cos[x])^((1 + a)/2)*(C[1]*HeunG[2, 1/4 - a, 1/2, 3/2, 1/2 - a, 
1, 1 + Cos[x]] + C[2]*(1 + Cos[x])^(1/2 + a)*Hypergeometric2F1[(1 + a)/2, (2 + a
)/2, 3/2 + a, Sin[x]^2]))/(1 + Cos[x])^(a/2)}}

Maple raw input

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

Maple raw output

[y(x) = _C1*hypergeom([-1/2*a, -1/2*a+1/2],[1/2-a],sin(x)^2)*sin(x)^(-a)+_C2*hyp
ergeom([1/2*a+1, 1/2*a+1/2],[3/2+a],sin(x)^2)*sin(x)^(1+a)]