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 = 1.22065 (sec), leaf count = 0 , could not solve
DSolve[-(a*(1 + a)*Csc[x]^2*y[x]) - Tan[x]*Derivative[1][y][x] + Derivative[2][y][x] == 0, y[x], x]
Maple ✓
cpu = 0.218 (sec), leaf count = 61
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}(-{\frac {a}{2}},-{\frac {a}{2}}+{\frac {1}{2}};\,{\frac {1}{2}}-a;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \left ( \sin \left ( x \right ) \right ) ^{-a}+{\it \_C2}\,{\mbox {$_2$F$_1$}({\frac {a}{2}}+1,{\frac {1}{2}}+{\frac {a}{2}};\,{\frac {3}{2}}+a;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \left ( \sin \left ( x \right ) \right ) ^{1+a} \right \} \] 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
DSolve[-(a*(1 + a)*Csc[x]^2*y[x]) - Tan[x]*Derivative[1][y][x] + Derivative[2][y][x] == 0, y[x], x]
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),'implicit')
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*hypergeom([1/2*a+1, 1/2+1/2*a],[3/2+a],sin(x)^2)*sin(x)^(1+a)