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Homework Help: How can I find the analytical solution for the system?

  1. Dec 29, 2014 #1
    • Member warned about posting with no effort shown
    1. The problem statement, all variables and given/known data

    dx(t)/dt = N0*sin(omega*t) * x(t) - ( N0*x^2 / k )

    Omega,N0 and k are positive .

    2. Relevant equations

    3. The attempt at a solution
    I tried to solve it using the Bernoulli equations but I could not get the last result.
  2. jcsd
  3. Dec 29, 2014 #2


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    Show us what happens with the Bernoulli substitution so we can see what your difficulty is.
  4. Dec 30, 2014 #3


    Staff: Mentor

    Here's the DE in a bit more readable form.
    $$ \frac{dx}{dt} = N_0x(\sin(\omega t) - \frac{x}{k})$$
    Last edited: Dec 30, 2014
  5. Dec 30, 2014 #4
    The result that I got is :

    exp( (cos(omega*t) * (-N0/omega)) / x(t) = (N0 / k) * ( integral (exp( (cos(omega*t)) * (-N0/omega) ) ) ) dt

    So I do not know how to find the integral when typically there is no specific solution for the integral of the Exponential function.
  6. Dec 30, 2014 #5


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    You may just have to express the result in terms of unevaluated integrals. Integrals of the form ##\int e^{\frac 1 \omega \cos(\omega t)}~dt## don't have elementary antiderivatives.
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