No definitive answer yet, but both equations are linear, if the other variable is treated as an unknown input function. One could try to exploit this somehow.

Thanks Maik. It's a good point and we'll certainly try. :)

Hi Davoud: Surely we already had the numerical solution. We just wonder if it is possible to get a closed form of the solution.

Not likely. You can say something about the path though, which may be useful for what you want:

e^t x = y - \frac{\alpha}{\lambda}\ln\left(y-\frac{\alpha}{\lambda}\right)

For a derivation, see the attachment.

I hope I did not make any errors in - signs and such. If so, let me know.

First Eliminate 1-x between two equations. Then by integration from two sides of ensuing equation you can find xexp(t)=y-(\alpha/\lambda)log(/\lambday-\alpha). I envisage this relation will help you to analysis the problem more efficiently.

Dear Yen: Have a look at the attached file

Dear Reza: Thanks for the detailed semi-solution. I obtained similar your final ODE for the defined $Y$ as well. Unfortunately the biological sensible parameter regime is neither $\alpha=0$ nor $\lambda=0$ (in our case $\alpha$ is of order 1E1, and $\lambda$ is roughly $1E-3$; though $\lambda$ is small, note that it couples with an exponential factor $e^t$ so it becomes the significant driving force at later time.)

Dear Gert van der Zwan: Yes we get the similar results, and your suggestion is extremely helpful for qualitative feature of the phase portrait in the $xe^t$-$y$ plane. We do on the other hand are interested in estimating the time it needed for the system to relaxed to the nbhd of its stable fixed point. If there exists no analytical solution, it is our plan to apply standard asymptotic analysis to get an estimation.

Dear Yen Ten Ling: I think it is unlikely that you find an analytical solution. If you substitute the result for z into eq. (6) you get an equation that looks rather unsolvable. I could not find anything in Gradshteyn/Ryzhik that even comes close. If you need relaxation times, an asymptotic analysis would indeed be the way to go.

Thanks, Gert. Will keep posting updates of our analysis here as a documentation.

PS sorry for misspelling your name.

Ah, never mind, no offence taken :)

I think this may help you.

You can try to simplify the equations by replacing x with z=e^t (1-x), then on the new set of the equations use the Laplace Transform of both sides (in both equations).

That does not help. The equations are not linear, and a term like z y leads to a convolution in Laplace space.

Dear Yen: Take a look at the attached file. I haven't any information bout the domains of variables x and y but I hope the attached series solution works.

I would like to handle the question, but I am unable to rewrite the equations; I do not understand the notation of the equations..

No worry! I found the solution to my difficulty in Reza Faal's answer!!!!!

The notation is in latex. Look in Reza's or my attached documents to see what the equations look like in more standard form.

For a given value of t, the analytic relationship between y and x can be written using the transcendental Lambert's W function. This may help you understand the phase space for the solutiions. Hope it helps.

Can you send it in more legible way, in word or pdf format?

[email protected]

Dear Slobodan: try copy-and-paste the equations to http://www.codecogs.com/latex/eqneditor.php

The only "on-the-spot" suggestion I have is to look for the solution of Gert van der Zwan and search for the solution of the nonlinear equation not in Ryzhik & Gradstein but in such ODE handbooks as those of Erich Kamke or, more recently, of Polyanin and Manzhirov (the one of nonlinear ODE because they have several - linear ODE, PDE, integral eqs.) You will be needing a bit of luck!

No analytic solutions, but could be easily explored in (alpha, lambda) numerically, e.g. with Mathematica

Hello David Gurarie (been a long time...)

As for the equations: interesting kinetic problem, and I wonder about the role of the factor e^t in the y-equation (assuming it's not a typo, I can't imagine a physical system that has such a drastic effect).

The only obvious solution (in addition to those described by Reza Faal) is the stationary solution (fixed point) $y=\alpha/\lambda, x=\alpha/(\alpha+1)$, and its generalization $y=\alpha/\lambda, x = \alpha/(\alpha+1)+C_1 exp(-(\alpha+1)t)$, where $C_1$ is an arbitrary constant.

@Eugen Terentjev: indeed it is not a typo. In fact the equation is devised to study early embryo development, where the DNA segments grows exponentially. One of the variable describes the length of the DNA and the other describe some preloaded proteins by the parents, which in turn bind to the DNA. It is, of course, only interested in the first few generations (where the embryo multiplies from 1 cell to about 2^12 cells). The question I posted, is purely out of my mathematical curiosity :)

This system has the following first integral:

y+e^t x=-\frac{\alpha}{\lambda}\ln |-\lambda y+\alpha|+c

where c is an integration constant determined from initial condition.

Another attribute of the system that is easily obtained is as follows:

let z(t) = y*exp(-lambda*t) .

then, dx/dt +(1 + alpha)*x +dz/dt +lambda*z = alpha. The steady state solution is :

(1 + alpha)*x +lambda *z = alpha.

I think Apostol did a great job. It's indeed the first integral which lowers the order down to one.

Granted conserved integrals and reduction, there'd be still no analytic solution for {x(t,...),y(t,...)}, as f-n f parameters.

A simple (under-appreciated) truth is that nowadays there is no need in analytic formulae. Mathematica ParametricNDSolve allows easy manipulation of solutions as f-ns of parameters, better than any messy algebra.

If you are looking for sort of numerical solution and you have access to MATLAB, use ODE45 command. You need to write your equations in state-space format and create a function for that. Then simply run ODE45 for that function and plot the results.

http://www.mathworks.co.uk/help/matlab/ref/ode45.html

You can use the methods of algebraic geometry to find solutions to the equilibria. By treating 't' as a fixed constant value, you can take a mesh in 't-space' and find the corresponding equilibria in terms of the variables and the parameters by using a Groebner bases technique. This will give you completely symbolic solutions for the equilibria.

Alternatively, you could consider taking a mesh in t, lambda, and alpha parameter-space (for values of interest in a specific region) and attain numerical solutions to the equilibria. At which point, you could then do some type of multivariate interpolation to approximate the solution curve.

Thanks all for the reply.

I think I've made my question clear: I'm looking for analytic solutions. Certainly I did not just type out the equations and toss it here; instead I've performed comprehensive analysis, at least to the best of my knowledge. Surely that included numerical analysis, asymptotic analysis, parameter exploration and some of the conserved quantity (first integrals) is known even way before analysis (because the model was "built" to analyze a system with a conserved quantity).

Since it has been posted for a while, I would suggest to cease further comments unless someone really has a closed form.