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

Context: Graduate
Thread starter asdf1
Start date Aug 13, 2005
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SUMMARY

The discussion focuses on solving a non-exact ordinary differential equation (ODE) represented by the equation (2x-4y+5)y' + x-2y+3=0. Participants explore shifting the coordinate axis and substituting variables to transform the equation into a separable form. The conversation highlights the identification of singular solutions, particularly the linear solution y = x/2 + 11/8, which serves as an envelope for the general solution. The analysis also delves into the behavior of the system as it approaches asymptotic limits and the implications of critical solutions in the context of eigenvalues.

PREREQUISITES

Understanding of ordinary differential equations (ODEs)
Familiarity with coordinate transformations and substitutions
Knowledge of singular solutions and their properties
Basic concepts of eigenvalues and eigenvectors in linear systems

NEXT STEPS

Study the method of solving non-exact ODEs using coordinate transformations
Learn about singular solutions and how to identify them analytically
Explore the relationship between linear systems and their eigenvalues
Investigate the asymptotic behavior of differential equations

USEFUL FOR

Mathematicians, physicists, and engineers dealing with differential equations, particularly those interested in the analysis of non-exact ODEs and their solutions.

Aug 16, 2005

saltydog

Science Advisor

Homework Helper

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.