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Equilibrium Points of DE

  1. Oct 27, 2013 #1

    FeDeX_LaTeX

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    Gold Member

    The problem statement, all variables and given/known data
    Find the equilibrium points of the system, determine their type and sketch the phase portrait.

    ##\frac{dx}{dt} = -3y + xy - 10, \frac{dy}{dt} = y^2 - x^2##

    The attempt at a solution

    Putting it together:

    ##\frac{dy}{dx} = \frac{y^2 - x^2}{-3y + xy - 10} \equiv \frac{Q(x,y)}{P(x,y)}##

    Here, we see that the horizontal nullclines are plotted along the line ##y = \pm x## and the vertical nullclines along the curve ##y = \frac{10}{x - 3}##.

    We form the Jacobian, i.e.

    J = ##\left(
    \begin{array}{cc}
    P_x & P_y \\
    Q_x & Q_y
    \end{array}
    \right)## = ##\left(
    \begin{array}{cc}
    y & x - 3 \\
    -2x & -2y
    \end{array}
    \right)##

    So ##-tr(J) = y## and ##det(J) = 2x^2 - 2y^2 - 3##.

    My question is, where do I go from here? Through using a differential equation plotter, I can see that the equilibrium points are a spiral source and spiral sink at (5,5) and (-2,-2) respectively. How does one deduce this from the Jacobian?
     
  2. jcsd
  3. Oct 27, 2013 #2

    FeDeX_LaTeX

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    Gold Member

    Never mind, I've overcomplicated it -- all I needed to do was solve that system of DEs for x and y (substituting x = y).

    The magic of the Homework board strikes again!
     
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