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
`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]
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)
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
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))