Solving a Non-Exact O.D.E. with Coordinate Axis Shift

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Discussion Overview

The discussion revolves around solving a non-exact ordinary differential equation (ODE) through various methods, including coordinate axis shifts and parameterization. Participants explore different approaches to find solutions, analyze the nature of singular solutions, and discuss the implications of their findings.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents the ODE and expresses difficulty in shifting the coordinate axis, noting that the equation is not exact.
  • Another participant reformulates the ODE into a separable form and suggests a substitution to facilitate solving.
  • A different participant anticipates a complex solution and proposes a parametric approach, introducing differential operators to eliminate variables.
  • Some participants question whether the ODE is indeed non-exact, providing calculations that suggest it might be exact.
  • One participant suggests educated guessing leads to simpler linear solutions, discussing the nature of singular solutions and their relationship to the general solution.
  • Another participant describes the behavior of the system asymptotically and proposes a method to analyze singular solutions based on asymptotic behavior.
  • Discussions arise regarding the conditions under which a solution can be considered singular and the implications of such solutions on the general solution.
  • Participants explore the relationship between different systems of equations and their dynamics, questioning how they relate to the original system.

Areas of Agreement / Disagreement

Participants express differing views on whether the ODE is exact or not, and there is no consensus on the nature of the singular solutions. The discussion remains unresolved regarding the classification of certain solutions and their implications.

Contextual Notes

Some participants mention limitations in their approaches, such as the dependence on specific initial conditions and the complexity of deriving singular solutions analytically without guessing.

Who May Find This Useful

This discussion may be of interest to those studying differential equations, particularly in the context of non-exact ODEs, singular solutions, and the application of parameterization in solving such equations.

  • #31
I just realized something: For this problem and the other one Asdf posted, the solution, using Laplace Transforms, can be obtained in three easy steps in Mathematica:

Code: 
alist = {u, v} /. 
    Solve[{s u == 2 u - 4 v + 5/s, s v - 1 == - u + 2 v - 3/s}, {u, v}]
x = InverseLaplaceTransform[alist[[1, 1]], s, t]
y = InverseLaplaceTransform[alist[[1, 2]], s, t]

I find that amazing! Granted, in general, I'd have to include two extra lines to first calculate the transform and this doesn't help one learn the math; I would not recommend this to anyone just learning the technique, but once learned, this provides an effective, concise means of approching the global behavior of these systems. :smile:
 

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