ODE
\[ a y(x) \csc ^2(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0805149 (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 = 0.345 (sec), leaf count = 132
\[ \left \{ y \left ( x \right ) ={1\sqrt [4]{2\,\cos \left ( 2\,x \right ) +2} \left ( {\frac {\cos \left ( 2\,x \right ) }{2}}-{\frac {1}{2}} \right ) ^{{\frac {1}{4}\sqrt {1-4\,a}}}\sqrt {-2\,\cos \left ( 2\,x \right ) +2} \left ( \sqrt {2\,\cos \left ( 2\,x \right ) +2}{\mbox {$_2$F$_1$}({\frac {1}{4}\sqrt {1-4\,a}}+{\frac {3}{4}},{\frac {1}{4}\sqrt {1-4\,a}}+{\frac {3}{4}};\,{\frac {3}{2}};\,{\frac {\cos \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})}{\it \_C2}+{\mbox {$_2$F$_1$}({\frac {1}{4}\sqrt {1-4\,a}}+{\frac {1}{4}},{\frac {1}{4}\sqrt {1-4\,a}}+{\frac {1}{4}};\,{\frac {1}{2}};\,{\frac {\cos \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})}{\it \_C1} \right ) {\frac {1}{\sqrt {\sin \left ( 2\,x \right ) }}}} \right \} \] 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),'implicit')
Maple raw output
y(x) = (2*cos(2*x)+2)^(1/4)*(1/2*cos(2*x)-1/2)^(1/4*(1-4*a)^(1/2))*(-2*cos(2*x)+2)^(1/2)*((2*cos(2*x)+2)^(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)*_C2+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)*_C1)/sin(2*x)^(1/2)