ODE
\[ \left (b^2-a^2\right ) y(x)+2 a \cot (a x) y'(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.1099 (sec), leaf count = 58
\[\left \{\left \{y(x)\to \frac {1}{2} e^{-\sqrt {-b^2} x} \csc (a x) \left (\frac {c_2 e^{2 \sqrt {-b^2} x}}{\sqrt {-b^2}}+2 c_1\right )\right \}\right \}\]
Maple ✓
cpu = 0.089 (sec), leaf count = 24
\[ \left \{ y \left ( x \right ) ={\frac {{\it \_C1}\,\sin \left ( bx \right ) +{\it \_C2}\,\cos \left ( bx \right ) }{\sin \left ( ax \right ) }} \right \} \] Mathematica raw input
DSolve[(-a^2 + b^2)*y[x] + 2*a*Cot[a*x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> ((2*C[1] + (E^(2*Sqrt[-b^2]*x)*C[2])/Sqrt[-b^2])*Csc[a*x])/(2*E^(Sqrt[-b^2]*x))}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+2*a*cot(a*x)*diff(y(x),x)+(-a^2+b^2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C1*sin(b*x)+_C2*cos(b*x))/sin(a*x)