##### 4.25.31 $$y(x) \left (a+b \sinh ^2(x)\right )+y''(x)=0$$

ODE
$y(x) \left (a+b \sinh ^2(x)\right )+y''(x)=0$ ODE Classiﬁcation

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 0.183235 (sec), leaf count = 53

$\left \{\left \{y(x)\to c_1 \text {MathieuC}\left [\frac {1}{2} (b-2 a),\frac {b}{4},i x\right ]-c_2 \text {MathieuS}\left [\frac {1}{2} (b-2 a),\frac {b}{4},i x\right ]\right \}\right \}$

Maple
cpu = 1.662 (sec), leaf count = 39

$\left [y \left (x \right ) = \textit {\_C1} \MathieuC \left (\frac {b}{2}-a , \frac {b}{4}, i x \right )+\textit {\_C2} \MathieuS \left (\frac {b}{2}-a , \frac {b}{4}, i x \right )\right ]$ Mathematica raw input

DSolve[(a + b*Sinh[x]^2)*y[x] + y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> C[1]*MathieuC[(-2*a + b)/2, b/4, I*x] - C[2]*MathieuS[(-2*a + b)/2, b/
4, I*x]}}

Maple raw input

dsolve(diff(diff(y(x),x),x)+(a+b*sinh(x)^2)*y(x) = 0, y(x))

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)]