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Thread: Intergration of Differential Equations

  1. #1
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    Intergration of Differential Equations

    Hey all, need help.
    How do you find k once you have intergrated 1/NdN=kdt?
    I've got to:
    N=Ae^kt
    And i'm told:
    120=Ae^20k

    Thanks
    Trev
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by TrevK View Post
    Hey all, need help.
    How do you find k once you have intergrated 1/NdN=kdt?
    I've got to:
    N=Ae^kt
    And i'm told:
    120=Ae^20k

    Thanks
    Trev
    what is your A?
    if A has a value, then

    $\displaystyle k = \frac{ ln \frac{120}{A}}{20} = \frac{ ln 120 - ln A}{20}$
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  3. #3
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    Hello, Trev!

    Did they give us an initial condition?


    $\displaystyle \frac{dN}{N} \:=\:k\,dt$ . . . and we are given: .$\displaystyle N(20) \:=\:120$

    We have: .$\displaystyle \frac{dN}{N} \;=\;k\,dt$

    . . Integrate: .$\displaystyle \ln N \;=\;kt + c$

    . . Then: .$\displaystyle N \;=\; e^{kt+c} \;=\;e^{kt}\cdot e^c\;=\; Ce^{kt} $


    Assume that, when $\displaystyle t = 0,\;N \:=\:N_o$

    . . Then we have: .$\displaystyle N_o \:=\: Ce^0\quad\Rightarrow\quad C \:=\:N_o$

    . . The function (so far) is: .$\displaystyle N \;=\;N_oe^{kt}$


    We are told that: .$\displaystyle N(20) \:=\:120$
    . . Then we have: .$\displaystyle 120 \;=\;N_oe^{20k}\quad\Rightarrow\quad e^{20k} \;=\;\frac{120}{N_o}$
    . . Take logs: .$\displaystyle \ln\left(e^{20k}\right) \;=\;\ln\left(\frac{120}{N_o}\right) \quad\Rightarrow\quad 20k(\ln e) \;=\; \ln\left(\frac{120}{N_o}\right)$
    . . Hence: .$\displaystyle k \;=\;\frac{1}{20}\ln\left(\frac{120}{N_o}\right) $


    Therefore: .$\displaystyle N(t) \;=\;N_oe^{\left[\frac{1}{20}\ln(\frac{120}{N_o})\right]t} $



    Edit: Darn, just realized that this identical to kalagota's solution.
    .
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  4. #4
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    Thanks very much!

    Thank you for your help.
    I understand the question better now but yea i'm still not certain on how to find the actual answer.

    Just for verification A was just a substitute for C. The Arbitrary Constant.
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  5. #5
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by TrevK View Post
    Thank you for your help.
    I understand the question better now but yea i'm still not certain on how to find the actual answer.

    Just for verification A was just a substitute for C. The Arbitrary Constant.
    yeah i know.. what i min was, what would be the value of A? anyways, soroban had given the complete solution..
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