##### 4.25.21 $$a y(x) \csc ^2(x)+y''(x)=0$$

ODE
$a y(x) \csc ^2(x)+y''(x)=0$ ODE Classiﬁcation

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 0.220312 (sec), leaf count = 61

$\left \{\left \{y(x)\to \sqrt [4]{-\sin ^2(x)} \left (c_1 P_{-\frac {1}{2}}^{\frac {1}{2} \sqrt {1-4 a}}(\cos (x))+c_2 Q_{-\frac {1}{2}}^{\frac {1}{2} \sqrt {1-4 a}}(\cos (x))\right )\right \}\right \}$

Maple
cpu = 1.709 (sec), leaf count = 165

$\left [y \left (x \right ) = \frac {\textit {\_C1} \left (2 \cos \left (2 x \right )+2\right )^{\frac {1}{4}} \hypergeom \left (\left [\frac {\sqrt {1-4 a}}{4}+\frac {1}{4}, \frac {\sqrt {1-4 a}}{4}+\frac {1}{4}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \sqrt {-2 \cos \left (2 x \right )+2}\, \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {\sqrt {1-4 a}}{4}}}{\sqrt {\sin \left (2 x \right )}}+\frac {\textit {\_C2} \left (2 \cos \left (2 x \right )+2\right )^{\frac {3}{4}} \hypergeom \left (\left [\frac {\sqrt {1-4 a}}{4}+\frac {3}{4}, \frac {\sqrt {1-4 a}}{4}+\frac {3}{4}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \sqrt {-2 \cos \left (2 x \right )+2}\, \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {\sqrt {1-4 a}}{4}}}{\sqrt {\sin \left (2 x \right )}}\right ]$ Mathematica raw input

DSolve[a*Csc[x]^2*y[x] + y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (C[1]*LegendreP[-1/2, Sqrt[1 - 4*a]/2, Cos[x]] + C[2]*LegendreQ[-1/2,
Sqrt[1 - 4*a]/2, Cos[x]])*(-Sin[x]^2)^(1/4)}}

Maple raw input

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

Maple raw output

[y(x) = _C1*(2*cos(2*x)+2)^(1/4)/sin(2*x)^(1/2)*hypergeom([1/4*(1-4*a)^(1/2)+1/4
, 1/4*(1-4*a)^(1/2)+1/4],[1/2],1/2*cos(2*x)+1/2)*(-2*cos(2*x)+2)^(1/2)*(1/2*cos(
2*x)-1/2)^(1/4*(1-4*a)^(1/2))+_C2*(2*cos(2*x)+2)^(3/4)/sin(2*x)^(1/2)*hypergeom(
[1/4*(1-4*a)^(1/2)+3/4, 1/4*(1-4*a)^(1/2)+3/4],[3/2],1/2*cos(2*x)+1/2)*(-2*cos(2
*x)+2)^(1/2)*(1/2*cos(2*x)-1/2)^(1/4*(1-4*a)^(1/2))]