2.4.67 Problem 64

Internal problem ID [8956]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 64
Date solved : Friday, February 21, 2025 at 09:00:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

\begin{align*} \frac {x y^{\prime \prime }}{-x^{2}+1}+y&=0 \end{align*}

Maple step by step solution

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
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 
   -> Whittaker 
      -> 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 
trying a solution in terms of MeijerG functions 
-> 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 differential order: 2; exact nonlinear 
   trying symmetries linear in x and y(x) 
   trying to convert to a linear ODE with constant coefficients 
   trying to convert to an ODE of Bessel type 
   -> trying reduction of order to Riccati 
      trying Riccati sub-methods: 
         -> trying a symmetry pattern of the form [F(x)*G(y), 0] 
         -> trying a symmetry pattern of the form [0, F(x)*G(y)] 
         -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
--- Trying Lie symmetry methods, 2nd order --- 
`, `-> Computing symmetries using: way = 3`[0, y]
 

Maple dsolve solution

Solving time : 0.212 (sec)
Leaf size : maple_leaf_size

dsolve(x/(-x^2+1)*diff(diff(y(x),x),x)+y(x) = 0,y(x),singsol=all)
 

\[ \text {No solution found} \]

✗Mathematica DSolve solution

Solving time : 0.0 (sec)
Leaf size : 0

DSolve[{x/(1-x^2)*D[y[x],{x,2}]+y[x]==0,{}},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✓Sympy solution

Solving time : 0.739 (sec)
Leaf size : 32

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2))/(1 - x**2) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Eq(y(x), C1*x*(-83*x**4/2880 + 11*x**3/144 + x**2/12 - x/2 + 1) + 
O(x**6))
 

\[ y{\left (x \right )} = C_{1} x \left (- \frac {83 x^{4}}{2880} + \frac {11 x^{3}}{144} + \frac {x^{2}}{12} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]