ODE
\[ a y(x) \tan ^2(x)+y''(x)-2 \cot (2 x) y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.763615 (sec), leaf count = 42
\[\left \{\left \{y(x)\to \cos ^{1-\sqrt {1-a}}(x) \left (c_2 \cos ^{2 \sqrt {1-a}}(x)+c_1\right )\right \}\right \}\]
Maple ✓
cpu = 0.881 (sec), leaf count = 35
\[\left [y \left (x \right ) = \textit {\_C1} \left (\cos ^{1+\sqrt {1-a}}\left (x \right )\right )+\textit {\_C2} \left (\cos ^{1-\sqrt {1-a}}\left (x \right )\right )\right ]\] Mathematica raw input
DSolve[a*Tan[x]^2*y[x] - 2*Cot[2*x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> Cos[x]^(1 - Sqrt[1 - a])*(C[1] + C[2]*Cos[x]^(2*Sqrt[1 - a]))}}
Maple raw input
dsolve(diff(diff(y(x),x),x)-2*cot(2*x)*diff(y(x),x)+a*y(x)*tan(x)^2 = 0, y(x))
Maple raw output
[y(x) = _C1*cos(x)^(1+(1-a)^(1/2))+_C2*cos(x)^(1-(1-a)^(1/2))]