ODE
\[ a y(x) \tan ^2\left (\frac {x}{2}\right )+y''(x)-\csc (x) y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0884163 (sec), leaf count = 43
\[\left \{\left \{y(x)\to c_1 \cos \left (2 \sqrt {a} \log \left (\cos \left (\frac {x}{2}\right )\right )\right )-c_2 \sin \left (2 \sqrt {a} \log \left (\cos \left (\frac {x}{2}\right )\right )\right )\right \}\right \}\]
Maple ✓
cpu = 22.558 (sec), leaf count = 31
\[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( 1+\cos \left ( x \right ) \right ) ^{-i\sqrt {a}}+{\it \_C2}\, \left ( 1+\cos \left ( x \right ) \right ) ^{i\sqrt {a}} \right \} \] Mathematica raw input
DSolve[a*Tan[x/2]^2*y[x] - Csc[x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Cos[2*Sqrt[a]*Log[Cos[x/2]]] - C[2]*Sin[2*Sqrt[a]*Log[Cos[x/2]]]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)-diff(y(x),x)*csc(x)+a*y(x)*tan(1/2*x)^2 = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*(1+cos(x))^(-I*a^(1/2))+_C2*(1+cos(x))^(I*a^(1/2))