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Math Help - DIfferential equations

  1. #1
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    DIfferential equations

    Please help these are calculus revision question for a final exam coming up soon.

    Solve each of the following differential equations to find the corresponding growth function. (show all necessary calculus)

    i. dQ/dt=kQ where k>0 t>= 0 Q(0)= Qo



    2. A koala population N is subject to a birth rate a and a deah rate of b+y p per head of population per annum. Ecologists have modelled the rate of change of this population over time using the following:
    dN/dt = (a-b)N-yN^2

    It is knowln that a = 0.2, b=0.04, y= 2x10^-4 and the initial population was 200.
    Determine if and when the population will have doubled in size.

    I have this question down to t= int(1/(0.16N-2*1-^-4N^2) dn. I can't seem to integrate it properly. If someone could integrate and point me in the right direction for the rest of the question that'd be great.


    Another derivitives question for a homework sheet.

    The ends of a 4m long water trough are equilateral triangles whose sides are a metre in length.
    If water is being pumped into the trough at a rate of 1.5m^3 / min find the rate at which the water level is rising when the depth is 30 cm.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by phillipvdw2001 View Post
    2. A koala population N is subject to a birth rate a and a deah rate of b+y p per head of population per annum. Ecologists have modelled the rate of change of this population over time using the following:
    dN/dt = (a-b)N-yN^2

    It is knowln that a = 0.2, b=0.04, y= 2x10^-4 and the initial population was 200.
    Determine if and when the population will have doubled in size.

    I have this question down to t= int(1/(0.16N-2*1-^-4N^2) dn. I can't seem to integrate it properly. If someone could integrate and point me in the right direction for the rest of the question that'd be great.
    t=\int_{200}^{400} \frac{1}{(a-b)N-2. 10^{-4}N^2} ~dN = \int_{200}^{400} \frac{1}{N[(a-b)-2. 10^{-4}N]} ~dN

    Now use partial fraction on the integrand to turn it into something like:

    <br />
\frac{A}{N}+\frac{B}{(a-b)-2. 10^{-4}N}<br />

    RonL
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  3. #3
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    Hello, phillipvdw2001!

    Solve the differential equation to find the corresponding growth function.

    . . \frac{dQ}{dt}\:=\:kQ . where k > 0,\;t \geq 0,\;Q(0) = Q_o
    Separate variables: . \frac{dQ}{Q} \:=\:k\,dt

    Integrate: . \ln Q \:=\:kt + c

    Solve for Q\!:\;\;Q \:=\:e^{kt+c} \:=\;e^{kt}\cdot e^c\quad\Rightarrow\quad Q(t)\:=\:Ce^{kt}

    Since Q(0) \,=\,Q_o, we have: . Q_o \:=\:Ce^0\quad\Rightarrow\quad C \,=\,Q_o

    Therefore: . Q(t) \:=\:Q_oe^{kt}

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