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 = 0.978557 (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 = 0.271 (sec), leaf count = 91
\[ \left \{ y \left ( x \right ) = \left ( \sin \left ( bx \right ) \right ) ^{-{\frac {a-b}{2\,b}}} \left ( {\it LegendreQ} \left ( {\frac {1}{2\,b} \left ( -b+\sqrt {{a}^{2}+4\,c} \right ) },{\frac {a-b}{2\,b}},\cos \left ( bx \right ) \right ) {\it \_C2}+{\it LegendreP} \left ( {\frac {1}{2\,b} \left ( -b+\sqrt {{a}^{2}+4\,c} \right ) },{\frac {a-b}{2\,b}},\cos \left ( bx \right ) \right ) {\it \_C1} \right ) \right \} \] 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),'implicit')
Maple raw output
y(x) = 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))*_C2+LegendreP(1/2/b*(-b+(a^2+4*c)^(1/2)),1/2*(a-b)/b,cos(b*x))*_C1)