ODE
\[ \left (1-a^2 \cos ^2(x)\right ) y''(x)+a^2 \sin (x) \cos (x) y'(x)+y(x) \left (\text {a0}+\text {a1} \cos ^2(x)\right )=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 17.3906 (sec), leaf count = 0 , could not solve
DSolve[(a0 + a1*Cos[x]^2)*y[x] + a^2*Cos[x]*Sin[x]*Derivative[1][y][x] + (1 - a^2*Cos[x]^2)*Derivative[2][y][x] == 0, y[x], x]
Maple ✓
cpu = 3.561 (sec), leaf count = 225
\[\left [y \left (x \right ) = \textit {\_C1} \sqrt {a^{2} \cos \left (2 x \right )+a^{2}-2}\, \mathit {HG}\left (\frac {1}{a^{2}}, \frac {a^{2}-\mathit {a0}}{4 a^{2}}, \frac {3 a +\sqrt {a^{2}-4 \mathit {a1}}}{4 a}, \frac {\sqrt {a^{2}-4 \mathit {a1}}\, a +a^{2}+2 \mathit {a1}}{2 a \left (a +\sqrt {a^{2}-4 \mathit {a1}}\right )}, \frac {1}{2}, \frac {1}{2}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )+\frac {\textit {\_C2} \left (2 \cos \left (2 x \right )+2\right )^{\frac {3}{4}} \left (-2 \cos \left (2 x \right )+2\right )^{\frac {1}{4}} \sqrt {a^{2} \cos \left (2 x \right )+a^{2}-2}\, \mathit {HG}\left (\frac {1}{a^{2}}, \frac {4 a^{2}-\mathit {a0} +1}{4 a^{2}}, \frac {5 a +\sqrt {a^{2}-4 \mathit {a1}}}{4 a}, \frac {\sqrt {a^{2}-4 \mathit {a1}}\, a +a^{2}+\mathit {a1}}{a \left (a +\sqrt {a^{2}-4 \mathit {a1}}\right )}, \frac {3}{2}, \frac {1}{2}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )}{\sqrt {\sin \left (2 x \right )}}\right ]\] Mathematica raw input
DSolve[(a0 + a1*Cos[x]^2)*y[x] + a^2*Cos[x]*Sin[x]*y'[x] + (1 - a^2*Cos[x]^2)*y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[(a0 + a1*Cos[x]^2)*y[x] + a^2*Cos[x]*Sin[x]*Derivative[1][y][x] + (1 - a^2*Cos[x]^2)*Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve((1-a^2*cos(x)^2)*diff(diff(y(x),x),x)+a^2*diff(y(x),x)*cos(x)*sin(x)+(a0+a1*cos(x)^2)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(a^2*cos(2*x)+a^2-2)^(1/2)*HeunG(1/a^2,1/4/a^2*(a^2-a0),1/4*(3*a+(a^2-4*a1)^(1/2))/a,1/2/a/(a+(a^2-4*a1)^(1/2))*((a^2-4*a1)^(1/2)*a+a^2+2*a1),1/2,1/2,1/2*cos(2*x)+1/2)+_C2*(2*cos(2*x)+2)^(3/4)*(-2*cos(2*x)+2)^(1/4)*(a^2*cos(2*x)+a^2-2)^(1/2)/sin(2*x)^(1/2)*HeunG(1/a^2,1/4*(4*a^2-a0+1)/a^2,1/4/a*(5*a+(a^2-4*a1)^(1/2)),1/a/(a+(a^2-4*a1)^(1/2))*((a^2-4*a1)^(1/2)*a+a^2+a1),3/2,1/2,1/2*cos(2*x)+1/2)]