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Math Help - Question on autonomous differential equation

  1. #1
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    Question on autonomous differential equation

    I have the differential equation:

    dx/dt=x2+5x-6
    I have to find the equilibria and solve them.

    Second I have to answer the question:
    Let x1(t) be its particular solution such that x1(t1)=-7 - for some time instant t1. Is it possible that x1(t)=2 for some t? Explain.

    To be honest, I have no clue about how to answer to the second question...

    I find the points by setting:
    0=x2+5x-6
    x=-6 or x=1

    to get the equilibrium points:
    df/dx=2x+5
    = 17 for x=-6 and 7 for x=1

    so both points are unstable.
    Is this true? - How can I answer the second question.

    ----
    Another quick question:
    Is the differential equation linear? Is it separable?
    x'=x2+et+ex

    No it's not linear because it contains the second power of x. It's not separable, because it can not be written as dx/h(x)=t*g(t)
    Is this the right answer?
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  2. #2
    Junior Member
    Joined
    Jan 2009
    Posts
    56

    Re: Question on autonomous differential equation

    For x=-6 you get df/dx = -7, so x=-6 is stable point.

    So I guess but I cannot recall (took ODE courses 3-4 years ago, a bit rusty) that if at some time our solution equals -7, then because x=-6 is a stable solution all solutions will be attracted to it, thus there cannot be a time t such that x_1(t)=2.
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