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Math Help - Systems of differential equations and equilibrium points...

  1. #1
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    Systems of differential equations and equilibrium points...

    Hi again guys, hope you can help with this simple question...

    I am up to systems of differential equations and their resulting equilibrium points.

    Now, when faced with finding their equilibrium points, is the a simple way of knowing how many points we will find or are looking for:

    I am currently only up to working on two differential equations. The ones am working on are also linear.

    I am guessing that because they both only linear, I will have 2x equilibrium points?
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  2. #2
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    Quote Originally Posted by MaverickUK82 View Post
    Hi again guys, hope you can help with this simple question...

    I am up to systems of differential equations and their resulting equilibrium points.

    Now, when faced with finding their equilibrium points, is the a simple way of knowing how many points we will find or are looking for:

    I am currently only up to working on two differential equations. The ones am working on are also linear.

    I am guessing that because they both only linear, I will have 2x equilibrium points?
    In general if you have the the system ODE's

    \frac{dx}{dt}=P(x,y)

    \frac{dy}{dt}=Q(x,y)

    This equilibrium points are the solutions to the system of equation

    \begin{cases}P(x,y)=0 \\ Q(x,y)=0 \end{cases}

    If you have a plane autonomous system the linear system in P and Q will have exactly one solution, and hence one equilibrium point.
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  3. #3
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    A system of linear equation has either a unique solution, an infinite number of solutions, or no solution.

    That is, a system of any number of linear differential equations has either a single equilibrium point, an infinite number of equilibrium points (forming a subspace), or none.
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    That's bad because i've found two equilibrium points for the system I'm working on. Have I explained it correctly.
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    Quote Originally Posted by MaverickUK82 View Post
    That's bad because i've found two equilibrium points for the system I'm working on. Have I explained it correctly.
    What is the system of ODE's
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    x'=-axy+bx
    y'=cxy-dy


    the x' and y' are supposed to be liebniz notation, i.e., an x with a single dot above it but I can't do it in latex...
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    Quote Originally Posted by MaverickUK82 View Post
    x'=-axy+bx
    y'=cxy-dy


    the x' and y' are supposed to be liebniz notation, i.e., an x with a single dot above it but I can't do it in latex...
    First off the equations are not linear! You have products of x and y.

    you can you \dot for time derivatives

    \begin{cases}\dot{x}=-axy+bx \\ \dot{y}=cxy-dy \end{cases}

    P(x,y)=axy+bx=x(ay+b) and Q(x,y)=cxy-dy=y(cx-d)

    and yes there are two critical points
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