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Math Help - 1st Order Differential Equations

  1. #1
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    1st Order Differential Equations

    A population of insects is allowed to grow in a experimental environment. The rate of increase of the population is proportional to the number, n, of insects, at any time t days after the start of the experiment. Regarding n and t as continuous variables, form a differential equation relating n and t, and solve it to show that n=Ae^kt, where A and K are constants. The net increases during the fourth and fifth days are 350 and 500 insects respectively. Determine the population at the beginning of the fourth say. Hence, or otherwise, determine the population at the beginning of the first day
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    Quote Originally Posted by guess
    A population of insects is allowed to grow in a experimental environment. The rate of increase of the population is proportional to the number, n, of insects, at any time t days after the start of the experiment. Regarding n and t as continuous variables, form a differential equation relating n and t, and solve it to show that n=Ae^kt, where A and K are constants. The net increases during the fourth and fifth days are 350 and 500 insects respectively. Determine the population at the beginning of the fourth say. Hence, or otherwise, determine the population at the beginning of the first day
    This is a classical differencial equation. It is a mathematical model based on growth-which is exponential. Let Q(t) be a function which show the population at any given time. By the rule of growth:
    \frac{dQ(t)}{dt}=kQ(t) where k is some constant. This is a differencial equation with seperable terms, thus,
    \frac{dQ(t)}{Q(t)}=kdt
    Integrating,
    \ln Q(t)+C_1=kt+C_2
    Thus,
    \ln Q(t)=kt+C_3 where C_3=C_2-C_1
    Q(t)=e^{kt+C_3}=Ce^{kt} where C=e^{C_3}
    Notice when t=0 then Q(t) is the initial amount (the starting populations) thus,
    A=Q(0)=Ce^{k\cdot 0}=Ce^0=C
    Thus, A=C, thus,
    Q(t)=Ae^{kt} for some k.
    ------------------
    Solve the problem, 4th day was 350 and 5th was 500.
    Substitute that infromation into the exponential equation,
    \left\{\begin{array}{cc}350=&Ae^{4k}\\500=&Ae^{5k}  \end{array}\right
    Divide the second equation by the first to get,
    \frac{Ae^{5k}}{Ae^{4k}}=\frac{500}{350}
    Thus,
    e^k\approx 1.43 solve for k,
    k=\ln (1.43)\approx .357
    Thus, A^{e^.357\dot 4}=350
    Thus, A\approx 4.17
    Thus, the formula is, 4.17e^{.357t}
    Last edited by ThePerfectHacker; March 14th 2006 at 06:15 AM.
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  3. #3
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    can you further solve the question because i still don't really understand
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  4. #4
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    So sorry, i never see the last part.. thanks a lot
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    However i still can't get the answer. The answer is: 817 for the population at the beginning of the fourth day and 280 for the population at the beginning of the first day
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    Quote Originally Posted by ThePerfectHacker
    This is a classical differencial equation. It is a mathematical model based on growth-which is exponential. Let Q(t) be a function which show the population at any given time. By the rule of growth:
    \frac{dQ(t)}{dt}=kQ(t) where k is some constant. This is a differencial equation with seperable terms, thus,
    \frac{dQ(t)}{Q(t)}=kdt
    Integrating,
    \ln Q(t)+C_1=kt+C_2
    Thus,
    \ln Q(t)=kt+C_3 where C_3=C_2-C_1
    Q(t)=e^{kt+C_3}=Ce^{kt} where C=e^{C_3}
    Notice when t=0 then Q(t) is the initial amount (the starting populations) thus,
    A=Q(0)=Ce^{k\cdot 0}=Ce^0=C
    Thus, A=C, thus,
    Q(t)=Ae^{kt} for some k.
    ------------------
    Hacker, you managed to make that hard work

    Observe:

    \frac{dQ(t)}{dt}=kQ(t)

    this is virtualy the defining relation for the exponential function, so
    obviously this has a particular integral:

    <br />
\mathcal{Q}(t)=e^{kt}<br />

    and so the general solution is:

    <br />
Q(t)=Ae^{kt}<br />

    for some constant A.

    RonL
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    Quote Originally Posted by ThePerfectHacker
    Solve the problem, 4th day was 350 and 5th was 500.
    Substitute that infromation into the exponential equation,
    \left\{\begin{array}{cc}350=&Ae^{4k}\\500=&Ae^{5k}  \end{array}\right
    No the increase on the fourth and fith days were 350 and 500 respectivly,
    so:

    <br />
350=Q(4)-Q(3)
    <br />
500=Q(5)-Q(4)<br />
,

    which should allow us to find A and k

    RonL
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    Quote Originally Posted by CaptainBlack
    No the increase on the fourth and fith days were 350 and 500 respectivly,
    so:

    <br />
350=Q(4)-Q(3)
    <br />
500=Q(5)-Q(4)<br />
,
    These equations expand to:

    <br />
350=A(e^{4k}-e^{3k})
    <br />
500=A(e^{5k}-e^{4k})<br />
,

    Dividing the secon of these by the first gives:

    <br />
\frac{500}{350}={e^{4k}(e^k-1) \over e^{3k}(e^k-1) }=e^k<br />

    so:

    <br />
k=\ln\left(\frac{500}{350}\right)\approx 0.356675<br />

    Now:

    <br />
A=\frac{350}{e^{4k}-e^{3k}} \approx 280.1165<br />

    So:

    <br />
Q(t)=280.1165 e^{0.356675t}<br />
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  9. #9
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    Thanks Bro.. you got it!!
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  10. #10
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    Quote Originally Posted by guess
    However i still can't get the answer. The answer is: 817 for the population at the beginning of the fourth day and 280 for the population at the beginning of the first day
    We have:

    <br />
Q(t)=280.1165 e^{0.356675t}<br />

    At the beginning of the fourth day the population is:

    <br />
Q(3)=816.666 \approx 817<br />
.

    At the beginning of the first day the population is:

    <br />
Q(0)=280.1165 \approx 280<br />
.

    RonL
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  11. #11
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    Sorry bout that, the phrase "... an increase of..." got me.
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