It is possible to find an exact solution (hopefully in "close form") to $$y''+\sqrt{|y|}\operatorname{sgn}(y)+\sqrt{|y'|}\operatorname{sgn}(y')=0, \,y'(0)=0,\,y(0)= 1/4$$? How?...
There exist a value of $t^*=\,?$ from which $y(t)=0,\,\forall t\geq t^*$?
PS: If I multiply the equation by $i$ the terms looks alike Hilbert Transforms... I wasn´t able to do something with this, but maybe it helps someone else.
**** 3rd Added later****
The main things I would like to know (in case that the solution is to hard to find):
i) How it behaves for $t<0$?
ii) Does it have a ending time $t_f$ from where $y(t)=0,\, t>t_f$?
iii) Does the solution $y(t)$ been a compact-supported function?
iv) Does the solution $y(t)$ been a bump function $\in C_c^{\infty}$?,
v) Does the solution $y(t)$ starts with a discontinuity?
vi) Does the solution $y(t)$ lives always in the reals $y(t) \in \mathbb{R}\,\forall t$?
vi) Is the solution unique?
This how the solution looks on Wolfram-Alpha.
**** 2nd Added later****
Following a comment and since I am interested in a solution with $y\geq 0\Rightarrow |y|\equiv y$, following the graph of Wolfram-Alpha it also happen than $y'\leq 0\Rightarrow |y'|\equiv -y'$, and also using that $\text{sgn}(y)\sqrt{|y|} \equiv \frac{y}{|y|}\sqrt{|y|}\equiv \frac{y}{\sqrt{|y|}}$, I could change the equation into another equation that I believe are not "formally" equivalent, but will give the same solution to the initial value problem: $$y'' +\sqrt{y}+i \sqrt{y'}=0, \,y'(0)=0,\,y(0)= 1/4$$ and in Wolfram-Alpha the plot looks similar, but I also get stack this time, maybe some knows how to solve this other differential equation.
Plese note this analysis is only valid for this case $y(0)=1/4$, since for higher initial values it can be seen that the solution do oscillations around $y=0$ so the assumption $y>0$ is not always hold.
***** 1st Added later *****
Since someone close the question because of lack of background, I will explain why I think is an interesting question.
Recently I have learned here on MSE that no non-constant real-valued and continuous ANALYTIC function can be of finite-duration (since its compact-supported in the time variable), and also that no finite-duration function could be a solution to a linear ordinary differential equation (Linear ODE) - because of uniqueness-related-issues of solutions at the beginning and at the end which becomes zero outside them, so, since everything I have seen in engineering, which is been modeled or through Linear ODE or through solutions than can be described as a Power Series (analytical), are only approximations, since NO FINITE-DURATION solution to a system could be described by them.
I have been looking for a Theory that described continuous-time finite-duration systems and its solutions (which now I know must be non-linear), but I can´t find yet any related theory.... All I have found yet on the web are abstract things I don´t understand (modern differential topology/geometry), but all they starts with assumptions of, or, with Linear Operators or Analytical Manifolds (which I already know can´t model a finite-duration solution), or with Smooth Manifolds or Smooth diffeomorphism, which only can be of finite-duration if at their support in the time-axis the function is zero, which is too restrictive (because, if it is of finite-duration, is of compact support in the time variable, and the only way a smooth function could be of compact support is through been a bump-function in the time axis, requiring that their values and also all their derivatives are zero at the beginning and at the end, to keep smoothness).
As example, the simple model of a particle that experience a elastic collision with a wall, if the position vs time function is continuous (so no "teleporting" is allowed), the derivative/speed will show a bounded "jump discontinuity", which will become a singularity in the acceleration profile, so smoothness is no a desirable requirement (is too restrictive).
Since finite-duration non-smooth phenomena is the most abundant kind of system in daily life, I was expecting to find a lot of math related on internet, but disappointingly, I have only found a few (four) papers working with continuous-time finite-duration differential equations (I take the question from one of them): 1, 2, 3 and 4.
From my side, I believe I have found a way to find solutions to segments of functions that are already solutions of initial value problems Linear ODE, this by taking their finite-duration Fourier Transform, without needing to take the convolution with the rectangular function that made the "cut" of the full-time-solution, and avoiding the problem of the discontinuities at the edges of the compact-supported domain, keeping true the Parseval's relation - and this scheme is also useful for finding the finite-duration Fourier transform of non-linear functions if I know beforehand the finite-duration Fourier transform of its first or its second derivative, jointly with the border conditions - I explained it here.... but unfortunately, It doesn´t work with finite-duration systems which are necessarily non-linear (but is an easy alternative to being taking convolutions).
I have already know that if the solution start and end at zero, if smooth, it is a bump function (for which I am not really interested, since it implies a self-emerging system), but they also are quite complicated, and the only differential equation I found from them is in here, which is non-linear and also time-delayed, but is defined from all $t \in \mathbb{R}$ even where its solution lives only in a compact-support, so it fulfill what I have learned so far.
From the mentioned papers, I understand now that there exists continuous non-linear systems with finite-duration solutions, which solutions are not unique, so I am trying to understand If every continuous non-linear system could support finite-duration solutions or not, and if not, How you can identify systems with finite-duration solutions?, and also, If there are restrictions imposed to the solutions because of been of finite duration, like example, if they can have unbounded derivative or not because of it (my intuition says they will have more restricted conditions because of causality issues since an infinite speed violate every possible constant-causality-speed model, but is just speculation)....
And many other questions rise, since non-linear systems don´t preserve the superposition principle, How is possible that Maxwell's wave equations solutions to be "right" if finite-duration phenomena are non-linear? Which spurious effects could be introducing modeling them with infinite-duration signals?, speaking in "physicists terms", I quite shocked with the "so little amount" of papers about finite-duration physics that could be found published on Google (maybe I am using wrong words, so any help with finding these theory will be of great help), and maybe huge restrictions could rise: as example, if a continuous function is of finite-duration, just because of these two restrictions, the function is also bounded, and it Fourier Transform is Analytic, which are a huge restrictions just because of been of finite-duration... maybe other issues will rise because of it finite extension.
But before, I am trying to understand these papers, which are a bit advanced for me (but not unattainable as the differential topology topics), and for this, I am trying to find the solution of the presented equation, which is supposed to be a continuous-time finite-duration differential equation, and when plotted, it looks like the half of a bumped-function similar to a Gaussian kernel, but I don´t know if its really becoming zero for every time that is bigger than an specific $t^*$, or if it behaves as vanishing at infinity as the Gaussian function does, so, not been a truly finite-duration solution. An also, I don´t know what is happening at the beginning: it is just starting at zero previous the beginning (implying a jump discontinuity at the start)? or have non-zero values outside its finite-duration (maybe, becoming a complex function)?, What is happening with the derivative at the discontinuity at the beginning?... all these question requiring to know the specific solution.
Hope its clear now, and hope you get interested as I am into these continuous-time finite-duration systems.
