My Math Forum solution to this differential equation

 Differential Equations Ordinary and Partial Differential Equations Math Forum

 May 5th, 2013, 11:47 AM #1 Newbie   Joined: May 2013 Posts: 3 Thanks: 0 solution to this differential equation Hi. Working in a research project, we came along this set of coupled differential equations: S1 = (p1 + q11 · Y1 + q12 · Y2) · (1 ? Y1 ? Y2) S2 = (p2 + q22 · Y2 + q21 · Y1) · (1 ? Y1 ? Y2)  where S1 and S2 are the derivative functions with respect to time of Y1 and Y2 respectively. p1, p2, q11, q22, q12 and q21 are all constants of the model. Are math skills are limited. Can you help us solve this set of equations (if possible)? Thanks a lot!!
May 5th, 2013, 03:24 PM   #2
Math Team

Joined: Dec 2006
From: Lexington, MA

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Re: solution to this differential equation

Hello, jazz_1969!

Quote:
 Working in a research project, we came along this set of coupled differential equations: [color=beige]. . [/color]S1 = (p1 + q11·Y1 + q12·Y2) · (1 ? Y1 ? Y2) [color=beige]. . [/color]S2 = (p2 + q22·Y2 + q21·Y1) · (1 ? Y1 ? Y2)  where S1 and S2 are the derivative functions with respect to time of Y1 and Y2 respectively. p1, p2, q11, q22, q12 and q21 are all constants of the model. Our math skills are limited. Can you help us solve this set of equations (if possible)?

All those subscripts are dazzling and blinding.

Let's make some substitutions:

[color=beige]. . [/color]$\begin{Bmatrix}x=&Y_1 \\ \\ \\ y=&Y_2 \end{Bmatrix} \;\;\;\;\begin{Bmatrix}a=&p_1=&d=&p_2\\ \\ \\ b=&q_{11}=&e=&q_{21}\\ c=&q_{12}=$

Then we have:[color=beige] .[/color]$\begin{Bmatrix}\frac{dx}{dt}=&(a\,+\,bx\,+\,cy)(1\,-\,x\,-\,y) \\ \\ \\ \frac{dy}{dt}=&(d\,+\,ex\,+\,fy)(1\,-\,x\,-\,y) \end{Bmatrix}=$

Maybe someone is willing to work on it now . . .

 May 5th, 2013, 08:52 PM #3 Newbie   Joined: May 2013 Posts: 3 Thanks: 0 Re: solution to this differential equation hehe... thank you soroban!

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