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Math Help - Separation of variables

  1. #1
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    Separation of variables

    Hi, I need help on this problem

    Suppose that a certain population has a growth rate that varies with time and that this population satisfies the differential equation

    dy/dt = (0.5 + sin t)y/5

    If y(0)=1, find or estimate the time at which the population has doubled.

    I think that this can be done with separation of variables. If we cross multiply then we get 5 dy (1/y) = dt (0.5 + sint)
    Then integrate:

    5 ln|y| = 0.5t - cost

    Does this seem correct?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by RB06 View Post
    Hi, I need help on this problem

    Suppose that a certain population has a growth rate that varies with time and that this population satisfies the differential equation

    dy/dt = (0.5 + sin t)y/5

    If y(0)=1, find or estimate the time at which the population has doubled.

    I think that this can be done with separation of variables. If we cross multiply then we get 5 dy (1/y) = dt (0.5 + sint)
    Then integrate:

    5 ln|y| = 0.5t - cost

    Does this seem correct?
    You seem ok so far except you forgot the arbitrary constant on the RHS. So,
    5 ln|y| = 0.5t - cost + C
    => ln|y| = 0.1t - 0.2cost + C
    => y = e^(0.1t - 0.2cost + C)
    Now y(0) = 1
    => 1 = e^(0.1(0) - 0.2cos(0) + C)
    => 0.1(0) - 0.2cos(0) + C = 0
    => - 0.2 + C = 0
    => C = 0.2

    so we have y = e^(0.1t - 0.2cost + 0.2)
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