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    10 Differential Equation Questions

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    Use separation of variables to solve to the ODE:

    N ̇=〖{k〗_+-k_- ln⁡〖(N)}N〗;k_+,k_- and t>0;N(0)= N_0>0.
    Hint: Use u = ln(N) for u-substitution.

    Substitute your solution into the differential equation and show that it is in fact the solution.

    Evaluate your solution for N(t) as t→∞ .

    For what values of N will N ̇=0?

    For what values of N_0 will
    N ̇(0)>0
    b) N ̇(0)<0

    Considering your answers to questions 3), 4) and 5), describe how N and N ̇ change from t=0 to t →∞ for cases a) and b) in question 5).
    Use separation of variables to solve to the
    ODE: dx/dt= α_1 x-α_2 x^2;t>0;x(0)= x_0>0
    You may use integral tables.
    Find the equilibrium points of the ODE.
    Evaluate the solution as t→∞
    What conclusions can you make relative to the answers to questions 2 and 3.

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    Solution Preview

    The equation is:
    If we set:
    The equation becomes:

    This is a separable equation:


    And since we get after applying initial condition:



    Note that:

    So indeed, solution (1.8) is indeed the solution of the differential equation.

    As we have (note that )
    As N approaches the steady state solution

    The equation is:
    The steady state solution occurs when
    Since is undefined at N=0 (though as ), the only other value for the steady state is:

    Which is ...

    Solution Summary

    The expert examines ten differential equation questions.