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 = 0.750669 (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 = 0.446 (sec), leaf count = 82
\[ \left \{ y \left ( x \right ) ={{\rm e}^{\sqrt {2}\sqrt {k} \left ( \cos \left ( x \right ) \right ) ^{2}}} \left ( \cos \left ( x \right ) {\it HeunC} \left ( 2\,\sqrt {2}\sqrt {k},{\frac {1}{2}},-{\frac {1}{2}},-{\frac {b}{2}},{\frac {3}{8}}-{\frac {a}{4}}+{\frac {b}{4}}-{\frac {k}{4}}, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) {\it \_C2}+{\it HeunC} \left ( 2\,\sqrt {2}\sqrt {k},-{\frac {1}{2}},-{\frac {1}{2}},-{\frac {b}{2}},{\frac {3}{8}}-{\frac {a}{4}}+{\frac {b}{4}}-{\frac {k}{4}}, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) {\it \_C1} \right ) \right \} \] 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),'implicit')
Maple raw output
y(x) = exp(2^(1/2)*k^(1/2)*cos(x)^2)*(cos(x)*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+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)*_C1)