4.4 problem 52

Internal problem ID [5813]
Internal file name [OUTPUT/5061_Sunday_June_05_2022_03_19_37_PM_32704329/index.tex]

Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section: Chapter 2. Linear homogeneous equations. Section 2.2 problems. page 95
Problem number: 52.
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.

\[ \boxed {y^{\prime \prime }-\cot \left (x \right ) y^{\prime }+\cos \left (x \right ) y=0} \]

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 symmetry of the form [xi=0, eta=F(x)] 
   checking if the LODE is missing y 
   -> Trying an equivalence, under non-integer power transformations, 
      to LODEs admitting Liouvillian solutions. 
      -> 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 
   -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
   <- Heun successful: received ODE is equivalent to the  HeunC  ODE, case  a <> 0, e <> 0, c = 0 
   Change of variables used: 
      [x = arcsin(t)] 
   Linear ODE actually solved: 
      t*(-t^2+1)^(1/2)*u(t)-diff(u(t),t)+(-t^3+t)*diff(diff(u(t),t),t) = 0 
<- change of variables successful`
 

✓ Solution by Maple

Time used: 2.0 (sec). Leaf size: 49

dsolve(diff(y(x),x$2)-cot(x)*diff(y(x),x)+cos(x)*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \left (1+\cos \left (x \right )\right ) \operatorname {HeunC}\left (0, 1, -1, -2, \frac {3}{2}, \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) \left (c_{1} +c_{2} \left (\int _{}^{\cos \left (x \right )}\frac {1}{\left (\textit {\_a} +1\right )^{2} \operatorname {HeunC}\left (0, 1, -1, -2, \frac {3}{2}, \frac {\textit {\_a}}{2}+\frac {1}{2}\right )^{2}}d \textit {\_a} \right )\right ) \]

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[y''[x]-Cot[x]*y'[x]+Cos[x]*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

Not solved