ODE
\[ y(x) (a+b \cos (2 x))+y''(x)=0 \] ODE Classification
[_ellipsoidal]
Book solution method
TO DO
Mathematica ✓
cpu = 0.025772 (sec), leaf count = 28
\[\left \{\left \{y(x)\to c_1 \text {MathieuC}\left [a,-\frac {b}{2},x\right ]+c_2 \text {MathieuS}\left [a,-\frac {b}{2},x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.396 (sec), leaf count = 21
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it MathieuC} \left ( a,-{\frac {b}{2}},x \right ) +{\it \_C2}\,{\it MathieuS} \left ( a,-{\frac {b}{2}},x \right ) \right \} \] Mathematica raw input
DSolve[(a + b*Cos[2*x])*y[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*MathieuC[a, -b/2, x] + C[2]*MathieuS[a, -b/2, x]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+(a+b*cos(2*x))*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*MathieuC(a,-1/2*b,x)+_C2*MathieuS(a,-1/2*b,x)