Internal problem ID [8601]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1021.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\left (a \left (\cosh ^{2}\relax (x )\right )+b \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 39
dsolve(diff(diff(y(x),x),x)+(a*cosh(x)^2+b)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \MathieuC \left (-\frac {a}{2}-b , \frac {a}{4}, i x \right )+c_{2} \MathieuS \left (-\frac {a}{2}-b , \frac {a}{4}, i x \right ) \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 40
DSolve[(b + a*Cos[x]^2)*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \text {MathieuC}\left [\frac {a}{2}+b,-\frac {a}{4},x\right ]+c_2 \text {MathieuS}\left [\frac {a}{2}+b,-\frac {a}{4},x\right ] \\ \end{align*}