ODE
\[ a \cot (b x) y'(x)+c y(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 4.01928 (sec), leaf count = 0 , could not solve
DSolve[c*y[x] + a*Cot[b*x]*Derivative[1][y][x] + Derivative[2][y][x] == 0, y[x], x]
Maple ✓
cpu = 1.131 (sec), leaf count = 105
\[\left [y \left (x \right ) = \textit {\_C1} \left (\sin ^{-\frac {a -b}{2 b}}\left (b x \right )\right ) \LegendreP \left (\frac {-b +\sqrt {a^{2}+4 c}}{2 b}, \frac {a -b}{2 b}, \cos \left (b x \right )\right )+\textit {\_C2} \left (\sin ^{-\frac {a -b}{2 b}}\left (b x \right )\right ) \LegendreQ \left (\frac {-b +\sqrt {a^{2}+4 c}}{2 b}, \frac {a -b}{2 b}, \cos \left (b x \right )\right )\right ]\] Mathematica raw input
DSolve[c*y[x] + a*Cot[b*x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[c*y[x] + a*Cot[b*x]*Derivative[1][y][x] + Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve(diff(diff(y(x),x),x)+a*cot(b*x)*diff(y(x),x)+c*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*sin(b*x)^(-1/2*(a-b)/b)*LegendreP(1/2/b*(-b+(a^2+4*c)^(1/2)),1/2*(a-b)/b,cos(b*x))+_C2*sin(b*x)^(-1/2*(a-b)/b)*LegendreQ(1/2/b*(-b+(a^2+4*c)^(1/2)),1/2*(a-b)/b,cos(b*x))]