ODE
\[ y(x) \left (\text {a0}+\text {a1} \cos ^2(x)+\text {a2} \csc ^2(x)\right )+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 1.41209 (sec), leaf count = 0 , could not solve
DSolve[(a0 + a1*Cos[x]^2 + a2*Csc[x]^2)*y[x] + Derivative[2][y][x] == 0, y[x], x]
Maple ✓
cpu = 0.37 (sec), leaf count = 80
\[ \left \{ y \left ( x \right ) = \left ( \sin \left ( x \right ) \right ) ^{{\frac {1}{2}}+{\frac {1}{2}\sqrt {-4\,{\it a2}+1}}} \left ( \cos \left ( x \right ) {\it HeunC} \left ( 0,{\frac {1}{2}},{\frac {1}{2}\sqrt {-4\,{\it a2}+1}},-{\frac {{\it a1}}{4}},{\frac {3}{8}}-{\frac {{\it a0}}{4}}-{\frac {{\it a2}}{4}}, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) {\it \_C2}+{\it HeunC} \left ( 0,-{\frac {1}{2}},{\frac {1}{2}\sqrt {-4\,{\it a2}+1}},-{\frac {{\it a1}}{4}},{\frac {3}{8}}-{\frac {{\it a0}}{4}}-{\frac {{\it a2}}{4}}, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) {\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[(a0 + a1*Cos[x]^2 + a2*Csc[x]^2)*y[x] + y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[(a0 + a1*Cos[x]^2 + a2*Csc[x]^2)*y[x] + Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve(diff(diff(y(x),x),x)+(a0+a1*cos(x)^2+a2*csc(x)^2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = sin(x)^(1/2+1/2*(-4*a2+1)^(1/2))*(cos(x)*HeunC(0,1/2,1/2*(-4*a2+1)^(1/2),-1/4*a1,3/8-1/4*a0-1/4*a2,cos(x)^2)*_C2+HeunC(0,-1/2,1/2*(-4*a2+1)^(1/2),-1/4*a1,3/8-1/4*a0-1/4*a2,cos(x)^2)*_C1)