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What are the solutions for $y(t)\cdot\left(y'(t) + a\right)=-b\sin(t)$?

  • It could be proben that there exists some solutions?
  • Are these solutions unique?
  • and obviously, which are these solutions? (Closed-form if possible)

Actually the question is simple, and no other info is required, but to encourage you to participate I will share a motivation, but this intro is not needed to find the answer (just in case you don´t want to read it - but is quite interesting in my opinion).


Motivation

I am trying to figure out if the equation of the classic damped non-linear pendulum $$y''(t) + ay'(t)+b\sin\left(y(t)\right)=0$$ admits finite-duration solutions ($\{a,\,b\}$ are constants): I believe it must to, because is the simplest realistic physical model I know, and experimentally it stops moving after some finite time.

But along with being the simplest case of "realistic" dynamical system, it is also known to don´t have any known close form solution, and every solution it has through approximations are "vanishing at infinite", don´t really being a "true" finite-duration functions.

Recently I have learned that every non-constant dynamical system, to be of finite-duration, it must be nonlinear, so its solutions won´t be analytical in the full domain... so everything I know as engineer through Taylor Series, Linear ODEs, and Power Series expansions are approximations: quite shocking at first... I know that equations are "models" tied to assumptions, but I never think before that every model I know are actually approximations since no finite-duration solution can be supported by them... neither dynamical systems with stands uniqueness of solutions will be an accurate model for finite-duration phenomena.

But also I know now that a finite-duration solution could been represented within its duration using known functions, as bump functions $\in C_c^\infty$, like $f(t) = e^{t^2/(t^2-1)},\,|t|\leq 1; \,(f(t)\text{ = 0, else})$ where the exponential function is defined to represent the solution only within its compact support.

And studying compact-supported function, I found a really interesting condition: if a function is continuous and compact-supported (as a finite-duration position-vs-time model where "teleportation" is forbidden), then it is also: (i) a bounded function, and (ii) its Fourier Transform is an analytic function, both actually quite huge restrictions, and I start to wonder if being of finite-duration could make same kind of restrictions to the maximum rate of change that dynamical system could achieve, like restrictions to fulfill causality.

So now, I am trying to see if a finite-duration alternative could be found for the solution of the nonlinear damped pendulum without using approximations (here for start from the very beginning of dynamical systems I know).

I have found recently two papers from the same author (V. T. Haimo / Vardia Haimo), that analyze finite-duration differential equations [1] and [2].

And in [1] on Theorem 2 point (i), it is said that, without losing generality by considering that the finishing time of the finite-duration solution happens at $t_F = 0$, for a second order dynamical system described by $\ddot{x}(t) = g(x(t),\dot{x}(t))$ such $g(0,0)=0$ (the system dynamics "die" at $t_F = 0$), with $g \in C^1(\mathbb{R}\setminus \{0\})$, then for the system to support finite-duration solutions, the following another differential equation must have solutions: $$q(z)\frac{dq(z)}{dz} = g(z,q(z)),\,q(0)=0$$

Honestly the papers are bit advanced to my mathematical skills, but if I didn´t made any mistakes, following the example given on the papers, the corresponding equation for finding if the nonlinear damped pendulum supports finite-duration solutions is: $$q(z)\cdot\left(q'(z) + a\right)=-b\sin(z)$$ where I have used and abuse of notation in the main question: the $y(t)$ of the main equation are not the same function $y(t)$ of the solutions of the nonlinear damped pendulum, neither their variables $t$ are the same (it just look more natural for asking as a differential equation dependent in time instead of an arbitrary variable $z$).

But I get stack here, since I don´t know how to figure out if the equation of the main question have solutions.

Hope you get interested in this as I am, unfortunately, I have found just a few papers on Google about continuous time finite-duration differential equations, and neither of them as a whole studied Theory, neither a Wikipedia page, so or it is a quite unexplored topic, or because of security reasons the publication are not published for general public (the papers where published in a corporation that works for the military, so it could be a feasible reason).

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  • $\begingroup$ Would a power series solution work? $\endgroup$ Jan 16 at 2:01
  • $\begingroup$ @TymaGaidash I don't think it will be possible at least in principle, since a power series in the variable time $t$, to be of finite-duration, will be equivalent of being compact-supported in this variable, and as others have point me in other questions, the only analytical function which compact-support is the zero function... but I don´t have enough background to prove or disprove them... if you can find the solution through a power series (maybe limiting its domain, which could be possible in principle, as is done for bump functions), it would be awesome. $\endgroup$
    – Joako
    Jan 16 at 2:10
  • $\begingroup$ @rrogers thanks for commenting. I don´t fully understand your answer: what you are explaining is for finding a numerical solution?, I am looking for the exact solution, so if your approach is on this path, I hope you can elaborate it in an answer. $\endgroup$
    – Joako
    Feb 23 at 2:30
  • $\begingroup$ The finite interval trick/tail needs the derivatives to be included in the tail xform. $\endgroup$
    – rrogers
    Feb 23 at 2:48
  • 1
    $\begingroup$ See math.stackexchange.com/questions/3791761/…. This is an ODE of Chini type. $\endgroup$
    – TheSimpliFire
    Feb 23 at 20:55

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