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Challenge Use Taylor series expansions to prove first-order Differential Equation

  1. Feb 6, 2012 #1
    Challenge!! Use Taylor series expansions to prove first-order Differential Equation

    Suppose dy/dt = f(y) has an equilibrium point at y = y0 and
    a) f'(y0) = 0, f''(y0) = 0, and f'''(y0) > 0: Is yo a source, a sink, or a node?
    b) f'(y0) = 0, f''(y0) = 0, and f'''(y0) < 0: Is yo a source, a sink, or a node?
    b) f'(y0) = 0 and f''(y0) > 0: Is yo a source, a sink, or a node?

    Also, prove the answer you pick is true for each part!

    I know that the answer for a) is source, b) sink, c) Node but I have no clue how to prove that is true.
    Can anyone help me to start the question?
     
  2. jcsd
  3. Feb 6, 2012 #2

    Mark44

    Staff: Mentor

    Re: Challenge!! Use Taylor series expansions to prove first-order Differential Equati

    How does your textbook define these terms: source, sink, node?
     
  4. Feb 6, 2012 #3
    Re: Challenge!! Use Taylor series expansions to prove first-order Differential Equati

    Let use an example to illustrate source, sink, and node
    For example, let assume the equilibrium points are y = -3 and y = 2. dy/dt < 0 fir -3 < y < 2, and dy/dt > 0 for y < -3 and y > 2. Given this information, y = -3 is a sink and y = 2 is a source.

    Node just mean if the left hand and the right hand of an equilibrium has the same side of derivative.
     
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