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

1. ## 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,
"

Shown below is what I'm trying to integrate to get x(t), which is shown above.

I'm in a loop here people
, any help would be greatly, greatly appreciated.

2. ## 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.

3. ## Re: Integrating Exponential Decay Model (In Dire need of help!)

ok I have,
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

4. ## Re: Integrating Exponential Decay Model (In Dire need of help!)

Originally Posted by Scook116
ok I have,
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 ....

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

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

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

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

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

$x(t)=\dfrac{u_0}{k}\left(1-e^{-kt}\right)-L$

5. ## 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.