ODE
\[ y(x) \left (a+b \cosh ^2(x)\right )+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0333032 (sec), leaf count = 53
\[\left \{\left \{y(x)\to c_1 \text {MathieuC}\left [-a-\frac {b}{2},\frac {b}{4},i x\right ]-c_2 \text {MathieuS}\left [-a-\frac {b}{2},\frac {b}{4},i x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.339 (sec), leaf count = 39
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it MathieuC} \left ( -{\frac {b}{2}}-a,{\frac {b}{4}},ix \right ) +{\it \_C2}\,{\it MathieuS} \left ( -{\frac {b}{2}}-a,{\frac {b}{4}},ix \right ) \right \} \] Mathematica raw input
DSolve[(a + b*Cosh[x]^2)*y[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*MathieuC[-a - b/2, b/4, I*x] - C[2]*MathieuS[-a - b/2, b/4, I*x]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+(a+b*cosh(x)^2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*MathieuC(-1/2*b-a,1/4*b,I*x)+_C2*MathieuS(-1/2*b-a,1/4*b,I*x)