##### 4.25.18 $$y(x) (a+b \cos (2 x)+k \cos (4 x))+y''(x)=0$$

ODE
$y(x) (a+b \cos (2 x)+k \cos (4 x))+y''(x)=0$ ODE Classiﬁcation

[_ellipsoidal]

Book solution method
TO DO

Mathematica
cpu = 6.183 (sec), leaf count = 0 , could not solve

DSolve[(a + b*Cos[2*x] + k*Cos[4*x])*y[x] + Derivative[2][y][x] == 0, y[x], x]

Maple
cpu = 2.864 (sec), leaf count = 93

$\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{\sqrt {2}\, \sqrt {k}\, \left (\cos ^{2}\left (x \right )\right )} \HeunC \left (2 \sqrt {2}\, \sqrt {k}, -\frac {1}{2}, -\frac {1}{2}, -\frac {b}{2}, \frac {3}{8}-\frac {a}{4}+\frac {b}{4}-\frac {k}{4}, \cos ^{2}\left (x \right )\right )+\textit {\_C2} \,{\mathrm e}^{\sqrt {2}\, \sqrt {k}\, \left (\cos ^{2}\left (x \right )\right )} \HeunC \left (2 \sqrt {2}\, \sqrt {k}, \frac {1}{2}, -\frac {1}{2}, -\frac {b}{2}, \frac {3}{8}-\frac {a}{4}+\frac {b}{4}-\frac {k}{4}, \cos ^{2}\left (x \right )\right ) \cos \left (x \right )\right ]$ Mathematica raw input

DSolve[(a + b*Cos[2*x] + k*Cos[4*x])*y[x] + y''[x] == 0,y[x],x]

Mathematica raw output

DSolve[(a + b*Cos[2*x] + k*Cos[4*x])*y[x] + Derivative[2][y][x] == 0, y[x], x]

Maple raw input

dsolve(diff(diff(y(x),x),x)+(a+b*cos(2*x)+k*cos(4*x))*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1*exp(2^(1/2)*k^(1/2)*cos(x)^2)*HeunC(2*2^(1/2)*k^(1/2),-1/2,-1/2,-1/2
*b,3/8-1/4*a+1/4*b-1/4*k,cos(x)^2)+_C2*exp(2^(1/2)*k^(1/2)*cos(x)^2)*HeunC(2*2^(
1/2)*k^(1/2),1/2,-1/2,-1/2*b,3/8-1/4*a+1/4*b-1/4*k,cos(x)^2)*cos(x)]