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

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The discussion revolves around solving a non-exact ordinary differential equation (ODE) using coordinate axis shifts and substitutions. The original equation is transformed into a separable form by substituting variables, leading to a more manageable solution. Participants explore the concept of singular solutions, noting that certain linear solutions serve as envelopes for the general solution, which can complicate the analysis. The conversation also touches on the relationship between different systems of equations and their eigenvalues, highlighting the dynamics of the solutions. Overall, the thread emphasizes the complexity of solving non-exact ODEs and the significance of understanding singular solutions in this context.
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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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