ODE
\[ y''(x)-\cot (2 x) y'(x)+2 y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.238621 (sec), leaf count = 64
\[\left \{\left \{y(x)\to -\frac {2}{3} c_2 \cos (2 x) \cos ^{\frac {3}{2}}(x) \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {7}{4};\cos ^2(x)\right )+\frac {1}{2} c_1 \cos (2 x)-2 c_2 \sin ^2(x)^{3/4} \cos ^{\frac {3}{2}}(x)\right \}\right \}\]
Maple ✓
cpu = 0.428 (sec), leaf count = 32
\[ \left \{ y \left ( x \right ) ={\it \_C2}\, \left ( \cos \left ( x \right ) \sin \left ( x \right ) \right ) ^{{\frac {3}{2}}}{\mbox {$_2$F$_1$}({\frac {1}{2}},2;\,{\frac {7}{4}};\, \left ( \sin \left ( x \right ) \right ) ^{2})}-2\, \left ( \cos \left ( x \right ) \right ) ^{2}{\it \_C1}+{\it \_C1} \right \} \] Mathematica raw input
DSolve[2*y[x] - Cot[2*x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[1]*Cos[2*x])/2 - (2*C[2]*Cos[x]^(3/2)*Cos[2*x]*Hypergeometric2F1[1/4, 3/4, 7/4, Cos[x]^2])/3 - 2*C[2]*Cos[x]^(3/2)*(Sin[x]^2)^(3/4)}}
Maple raw input
dsolve(diff(diff(y(x),x),x)-cot(2*x)*diff(y(x),x)+2*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C2*(cos(x)*sin(x))^(3/2)*hypergeom([1/2, 2],[7/4],sin(x)^2)-2*cos(x)^2*_C1+_C1