Wikipedia:Reference desk/Archives/Mathematics/2020 August 24

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August 24

what is the point with the continuation of solution of an ode?

Hi there, There is a theorem in ODE that Consider the ODE y 0 = f(x, y) where f is continuous in the plane and locally Lipschitz in y. Suppose y : (x0 − ε, x0 + ε) → R is an integral curve of the ODE passing through (x0, y0). Let I = (xL, xR) be the maximal possible open interval to which our ODE admits an extension (which may be unbounded in either direction, i.e. we allow xL to be −∞ and xR to be +∞). The principle of continuation says that there are two possibilities: either
1R. The integral curve can be extended infinitely far to the right,
2R. There exists a point xR such that limx→x − R y(x) = ±∞, i.e. the ODE blows up in finite time. A similar dichotomy holds when attempting to extend the integral curve to the left.

My question is: isn't it a null statement? what else can happen? I really don't get it.--Exx8 (talk) 19:04, 24 August 2020 (UTC)

Given that under the given conditions (or so I think) any extension is unique on its domain – which is not true for all ODE's – this just an enumeration of the possibilities. It is similar to the statement that either the universe will exist forever, or cease to exist at some point in the future. (I am not sure all cosmologists will agree, but interpret this in a naive theory of time). Another way of saying this is that, as long as the curve remains continuous (as a function of x) it can be extended, but this game is not guaranteed to keep making progress: you may run into an obstacle, which then takes the form of the curve "blowing up" – other types of discontinuity cannot happen. I am not perfectly sure that the formulation of possibility 2R is completely correct; I wonder if it shouldn't be $\lim _{x\rightarrow x_{R}}|y(x)|=\infty$.  --Lambiam 20:18, 24 August 2020 (UTC)

I think this just means a solution exists and is unique, unless it hits a singularity. See Picard–Lindelöf theorem. This is not true of PDE, which might have no solutions, multiple solutions, etc. 2601:648:8202:96B0:0:0:0:DDAF (talk) 04:39, 25 August 2020 (UTC)

Well, it is not a null statement, as one has to rule out some situations. In principle, a solution $u(t)$ may fail to be extendable even if it is bounded because it oscillates like crazy, like $\sin(1/t)$ at $t=0$, or it may be unbounded and fail to go to $+\infty$, like $(1/t)\sin(1/t)$ at $t=0$. Or, even more subtle situation, even if $u(t)$ does have a limit $\lim _{t\to x_{R}}u(t)=y$, it is not completely obvious that $u(t)$ may be extended on $(x_{L},x_{R}+\epsilon )$ by gluing to it on the right a local solution $v(t)$ of the initial value problem with $v(x_{R})=y$: why this extended function is differentiable and satisfies the ODE at $t=x_{R}$?pma 11:59, 25 August 2020 (UTC)

The first example violates the condition of Lipschitz continuity. I'll think about the second tomorrow as I'm too sleepy rn. 2601:648:8202:96B0:0:0:0:DDAF (talk) 08:42, 27 August 2020 (UTC)

It is not entirely clear what the meaning of "null statement" in the question is and what it refers to. Consider the following theorem:

Let a be a real number. Suppose a ≠ 0. The principle of discernment says that there are then two possibilities: either

1. a < 0, or
2. a > 0.

Under the conditions of the theorem, these are indeed the only possibilities. The theorem itself is not a vacuous truth, because the antecedent can be satisfied. The conclusion of the theorem is not a tautology; the condition of the theorem is necessary. Then, of course, pointing the latter out by giving an instantiation that falsifies the conclusion (like pma did) will necessarily mean that the condition of the theorem is violated.  --Lambiam 08:15, 28 August 2020 (UTC)