ODE
\[ -2 \csc (2 x) \left (1-a \sin ^2(x)\right ) y'(x)+b y(x) \tan ^2(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.921086 (sec), leaf count = 48
\[\left \{\left \{y(x)\to \cos ^{\frac {1}{2} \left (a-\sqrt {a^2-4 b}\right )}(x) \left (c_2 \cos ^{\sqrt {a^2-4 b}}(x)+c_1\right )\right \}\right \}\]
Maple ✓
cpu = 0.754 (sec), leaf count = 45
\[\left [y \left (x \right ) = \textit {\_C1} \left (\cos ^{\frac {a}{2}+\frac {\sqrt {a^{2}-4 b}}{2}}\left (x \right )\right )+\textit {\_C2} \left (\cos ^{\frac {a}{2}-\frac {\sqrt {a^{2}-4 b}}{2}}\left (x \right )\right )\right ]\] Mathematica raw input
DSolve[b*Tan[x]^2*y[x] - 2*Csc[2*x]*(1 - a*Sin[x]^2)*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> Cos[x]^((a - Sqrt[a^2 - 4*b])/2)*(C[1] + C[2]*Cos[x]^Sqrt[a^2 - 4*b])}}
Maple raw input
dsolve(diff(diff(y(x),x),x)-2*diff(y(x),x)*csc(2*x)*(1-a*sin(x)^2)+b*y(x)*tan(x)^2 = 0, y(x))
Maple raw output
[y(x) = _C1*cos(x)^(1/2*a+1/2*(a^2-4*b)^(1/2))+_C2*cos(x)^(1/2*a-1/2*(a^2-4*b)^(1/2))]