Internal problem ID [9354]
Internal file name [OUTPUT/8290_Monday_June_06_2022_02_38_51_AM_93890807/index.tex
]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\left (a \cosh \left (x \right )^{2}+b \right ) y=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius Equivalence transformation and function parameters: {z = 1/2*t+1/2}, {kappa = -4-4*b, mu = 4*a} <- Equivalence to the rational form of Mathieu ODE successful <- Mathieu successful <- special function solution successful Change of variables used: [x = 1/2*arccosh(t)] Linear ODE actually solved: (a*t+a+2*b)*u(t)+8*t*diff(u(t),t)+(8*t^2-8)*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.562 (sec). Leaf size: 39
dsolve(diff(diff(y(x),x),x)+(a*cosh(x)^2+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {MathieuC}\left (-\frac {a}{2}-b , \frac {a}{4}, i x \right )+c_{2} \operatorname {MathieuS}\left (-\frac {a}{2}-b , \frac {a}{4}, i x \right ) \]
✓ Solution by Mathematica
Time used: 0.052 (sec). Leaf size: 40
DSolve[(b + a*Cos[x]^2)*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \text {MathieuC}\left [\frac {a}{2}+b,-\frac {a}{4},x\right ]+c_2 \text {MathieuS}\left [\frac {a}{2}+b,-\frac {a}{4},x\right ] \]