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Math Help - Integrating Exponential Decay Model (In Dire need of help!)

  1. #1
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    Integrating Exponential Decay Model (In Dire need of help!)

    I'm having trouble on a question in my calculus project about "Landing an airliner". It is not very complex and I understand everything but this problem.
    They even give me the solution to the problem, just need to figure out how they got there.

    "Integrate the relation u(t)=dx/dt and use the condition x(0)=-L to show that,
    Integrating Exponential Decay Model (In Dire need of help!)-12a.png"

    Shown below is what I'm trying to integrate to get x(t), which is shown above.
    Integrating Exponential Decay Model (In Dire need of help!)-12.png


    I'm in a loop here people
    , any help would be greatly, greatly appreciated.
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  2. #2
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    Re: Integrating Exponential Decay Model (In Dire need of help!)

    $\dfrac{dx}{dt}=u_0 e^{-kt}$

    $dx = u_0 e^{-kt}dt$

    $x(t)=\displaystyle{\int }u_0 e^{-kt}dt$

    do this integration, include a constant of integration C, and solve for C given $x(0)=-L$

    you'll find the formula you have pops right out.
    Thanks from Scook116
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  3. #3
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    Re: Integrating Exponential Decay Model (In Dire need of help!)

    ok I have, Integrating Exponential Decay Model (In Dire need of help!)-k.png
    But i dont see how the answer can include (1-e^-kt)
    I've been at this for days, I cant see it im not sure why... The worst part is I know it's right in front of me
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  4. #4
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    Re: Integrating Exponential Decay Model (In Dire need of help!)

    Quote Originally Posted by Scook116 View Post
    ok I have, Click image for larger version. 

Name:	k.png 
Views:	8 
Size:	1.4 KB 
ID:	30290
    But i dont see how the answer can include (1-e^-kt)
    I've been at this for days, I cant see it im not sure why... The worst part is I know it's right in front of me
    hokay ....

    $x(t)=\displaystyle{\int }u_0 e^{-kt}dt$

    $x(t)=\dfrac{-u_0}{k}e^{-kt}+C$

    $x(0)=\dfrac{-u_0}{k}+C=-L$

    $C=\dfrac{u_0}{k}-L$

    $x(t)=\dfrac{-u_0}{k}e^{-kt}+\left(\dfrac{u_0}{k}-L\right)=$

    $x(t)=\dfrac{u_0}{k}\left(1-e^{-kt}\right)-L$
    Thanks from Scook116
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  5. #5
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    Re: Integrating Exponential Decay Model (In Dire need of help!)

    It was that 5th step that was gettin' me, but I see it....
    You sir, have the patience of a god.

    You helped me a great deal,
    Thank you.
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