4.27.19 \(y''(x)-\cot (2 x) y'(x)+2 y(x)=0\)

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 = 0.292059 (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 = 0.913 (sec), leaf count = 34

\[\left [y \left (x \right ) = \textit {\_C1} \left (-2 \left (\cos ^{2}\left (x \right )\right )+1\right )+\textit {\_C2} \left (\cos \left (x \right ) \sin \left (x \right )\right )^{\frac {3}{2}} \hypergeom \left (\left [\frac {1}{2}, 2\right ], \left [\frac {7}{4}\right ], \sin ^{2}\left (x \right )\right )\right ]\] 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))

Maple raw output

[y(x) = _C1*(-2*cos(x)^2+1)+_C2*(cos(x)*sin(x))^(3/2)*hypergeom([1/2, 2],[7/4],s
in(x)^2)]