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Population Dynamics how to solve a particular ODE

  1. Feb 20, 2013 #1
    1. The problem statement, all variables and given/known data

    I have a population problem where:

    [itex]\frac{dy}{dt}[/itex] = ay - [itex]by^{2}[/itex]-[itex]\frac{c*y^{3}}{d+y^{3}}[/itex]

    I need to find an expression for y(t). I'm not looking for the answer, just some advice/ helpful hints.

    Thank you.

    2. Relevant equations



    3. The attempt at a solution

    I know that the integrating factor method is out of the question, as is separation of variables. Bernoulli's equation will also not work. All of these require forms that are different than what is written above.
     
  2. jcsd
  3. Feb 21, 2013 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    Getting t in terms of y is not too hard, but getting y in terms of t is horrible. If you write
    [tex]F(y) \equiv \int\frac{dy}{f(y)} = \int dt,\\
    f(y) = a y - b y^2 - \frac{c y^3}{d + y^3},
    [/tex]
    the y-integral is doable. Using 'r' instead of 'd' (because 'd' is a reserved symbol), Maple gets the y-integral as

    F(y) = 1/2*a*x^2-1/3*b*x^3-c*x+1/3*c*r^(1/3)*ln(x+r^(1/3))
    -1/6*c*r^(1/3)*ln(x^2-x*r^(1/3)+r^(2/3))
    +1/3*c*r^(1/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/r^(1/3)*x-1))

    where Maple writes u/v*w to mean (u/v)*w. So, F(y) = t+C is a "solution", but inverting F to get y(t) is probably only doable numerically.
     
  4. Feb 21, 2013 #3
    Ah, I see. Separation of variables is completely doable! I don't know what I was thinking. Thank you so much.
     
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