3.22 problem 1022

Internal problem ID [9355]
Internal file name [OUTPUT/8291_Monday_June_06_2022_02_39_01_AM_20923796/index.tex]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1022.
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "unknown"

Maple gives the following as the ode type

[_ellipsoidal]

Unable to solve or complete the solution.

\[ \boxed {y^{\prime \prime }+\left (a \cos \left (2 x \right )+b \right ) y=0} \]

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-4*a, mu = -8*a} 
         <- Equivalence to the rational form of Mathieu ODE successful 
      <- Mathieu successful 
   <- special function solution successful 
   Change of variables used: 
      [x = 1/2*arccos(t)] 
   Linear ODE actually solved: 
      (a*t+b)*u(t)-4*t*diff(u(t),t)+(-4*t^2+4)*diff(diff(u(t),t),t) = 0 
<- change of variables successful`
 

✓ Solution by Maple

Time used: 0.61 (sec). Leaf size: 21

dsolve(diff(diff(y(x),x),x)+(a*cos(2*x)+b)*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = c_{1} \operatorname {MathieuC}\left (b , -\frac {a}{2}, x\right )+c_{2} \operatorname {MathieuS}\left (b , -\frac {a}{2}, x\right ) \]

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 28

DSolve[(b + a*Cos[2*x])*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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