Talk:Ordinary differential equation

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Removal of content[edit]

I deleted the section Linear equations rather than wasting time cleaning it up, since everything is covered in far more detial and in better presentation in the main article Linear differential equation.

Furthermore the subsections within that section: Fundamental systems for homogeneous equations with constant coefficients and General Case, had zero references and were not exactly easy for a reader to follow anyway... they are no more. F = q(E+v×B) ⇄ ∑_ic_i 09:34, 31 May 2012 (UTC)

There was a link to the section Linear ordinary differential equations in the article Stiff equation. I just now updated the link to point to the section Reduction of order instead, as this section describes not only the reduction of order but also the vector representation of such a system. Please consider leaving the existing section in place so that the vector representation will still be described in a convenient place. If the vector representation is deleted, it should be moved somewhere and documented in the talk pages for its source and destination pages. — Anita5192 (talk) 17:48, 31 May 2012 (UTC)

Of course I'll leave that section alone - thats relavent! My only implications were that the theory of linear equations should be kept in the other article (which is also in bad shape in places.....), rather than repeating too much here. Thank you for feedback - appreciated. =) F = q(E+v×B) ⇄ ∑_ic_i 23:27, 31 May 2012 (UTC)

Global uniqueness and maximum domain of solution[edit]

It would be nice to have a counter-example with domain ℝ\{x_0 + 1/y_0}, that satisfies the initial condition, but has a different definition on the other interval. Soulpa7ch (talk) 18:49, 17 September 2012 (UTC)

Error![edit]

f(x, y) = y^2 is *not* Lipschitz continuous. Please rectify or make clear (that you mean "locally Lipschitz"). — Preceding unsigned comment added by 2607:4000:200:12:21A:92FF:FE83:373 (talk) 23:24, 21 January 2013 (UTC)

Argument of function belongs in numerator[edit]

In the Background section, I moved the argument of x(t) back into the numerator of the derivative because that is where it commonly is placed in textbooks. See, e.g., Simmons, George F. Differential Equations with Applications and Historical Notes. p.123. — Anita5192 (talk) 20:32, 6 November 2014 (UTC)

Proposed merge with Strang splitting[edit]

seems to be one of the possible solutions Shrikanthv (talk) 10:53, 3 December 2014 (UTC)

Operator splitting is used for dimensional splitting of Partial differential equations as well, which has (almost 😉) nothing to do with ODEs. Merging strang splitting here thus doesn't make sense. I'd support merging strang splitting into a general article on splitting methods though. -- Pberndt (talk) 08:54, 19 April 2016 (UTC)

More rigor in the definitions of $n$ th order ODEs[edit]

In the definition of an $n$ th order linear ODE $\left(y^{(n)}=\sum _{i=0}^{n-1}a_{i}(x)y^{(i)}+r(x)\ \forall x\in I\right)$ , all the article says is that the $a_{i}$ and $r$ are continuous. It doesn't even say $\forall x\in I$ at the end of the equation. I think we should give the domains and codomains of all the functions in the equation, i.e. we should say that $a_{0},a_{1},...,a_{n-1},r:I\to \mathbb {R}$ are continuous and $y:I\to \mathbb {R}$ is $n$ times differentiable (from which it follows from the equation that $y$ is actually $n$ times continuously differentiable).

In the definition of a general $n$ th order ODE (implicit form $F(x,y,y',...,y^{(n)})=0\ \forall x\in I)$ and explicit form $y^{(n)}=F(x,y,y',...,y^{(n-1)})\ \forall x\in I)$ ), the article says even less. Again there is no $\forall x\in I$ , and it doesn't say that $y:I\to \mathbb {R}$ should be $n$ times differentiable. The most problematic part however is that nothing at all is said about $F$ . In the implicit case, its domain and codomain are given by $F:I\times U\to \mathbb {R}$ where $U\subseteq \mathbb {R} ^{n+1}$ , and in the explicit case it is $F:I\times U\to \mathbb {R}$ where $U\subseteq \mathbb {R} ^{n}$ .

In the same way that we demand that $a_{0},a_{1},...,a_{n-1},r:I\to \mathbb {R}$ are continuous in the definition of a linear ODE, are there any agreed upon assumptions on $F$ in the definition of a general ODE? Later in the article in the 'Local existence and uniqueness theorem simplified' section, when talking about the first order system ${\vec {y}}'={\vec {F}}(x,{\vec {y}})$ (where yet again all the terms are not fully explained, or even made clear that they are vector quantities), it says that we should have ${\vec {F}}$ and ${\frac {\partial {\vec {F}}}{\partial {\vec {y}}}}$ continuous in some vicinity of the initial condition in order to guarantee the existence and uniqueness of a solution. Is there a similar result which applies to the general $n$ th order ODE $F(x,y,y',...,y^{(n)})=0\ \forall x\in I$ or $y^{(n)}=F(x,y,y',...,y^{(n-1)})\ \forall x\in I$ ? (I could probably figure it out by converting them into first order systems and reverse engineering, but I'll see what you guys say first).

So in summary, 1) do you guys think that we should modify the definitions of linear and general ODEs by explaining all the terms in the equations more fully, and 2) are there agreed upon assumptions on $F$ in the literature in the definition of a general (implicit and explicit) $n$ th order ODE? At the moment, having $F$ being identically $1$ for an implicit $n$ th order ODE satisfies the definition given in the article. — Preceding unsigned comment added by Joel Brennan (talk • contribs) 14:56, 21 April 2018 (UTC)

I agree with your first point, although for the sake of compactness I would include the assumptions valid for all the definitions in advance rather than in every definition. I'm sorry that I cannot answer to your second point. I also think that, after the general definition it would not be correct saying "There are further classifications". It could be said "general definitions for particular types of differential equations". However, i find it redundant and in some cases wrong what is presented. For instance, the type "homogeneous" refers only to linear equations so it is misleading this presentation. I would perhaps delete all this and just say that there are different particular types of equations, and provide the links to the correspondent articles, which I think are ok. What do you think? Conjugado (talk) 19:11, 2 January 2019 (UTC)

Software section[edit]

The software section doesn't actually include ODE solver software, but instead links to languages which have submodules that can solve ODEs. I tried to improve the software section by linking directly to one of the more popular submodules, DifferentialEquations.jl solvers, but MrOllie keeps deleting the reference. Should this section instead be renamed to "Languages with ODE Solver Software"? If that's the case, languages like Python, R, etc. should probably be removed, since their ODE solvers are wrappers for the C++ and Fortran implementations of methods like dopri5, dop853, and lsode. — Preceding unsigned comment added by 71.232.17.207 (talk) 12:26, 18 October 2019 (UTC)

The section should include only entries with some demonstrated notability in the form of a preexisting Wikipedia article. See WP:WTAF - MrOllie (talk) 12:28, 18 October 2019 (UTC)

So notability of an ODE solver is not defined in terms of users, Github metrics, citing articles, etc., but instead in terms of whether there's a preexisting Wikipedia article? It's fine if it's consistent, but that seems like an odd criteria. — Preceding unsigned comment added by 71.232.17.207 (talk) 13:08, 18 October 2019 (UTC)

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