Expert: Paul Klarreich - 5/9/2007

Question
Phoebe is returning to earth in her spaceship when she detects an oxygen tank leak. She knows that the rate of change of the pressure is directly proportional to the pressur of the remaining oxygen.
Use P for pressure and t for time. write a differential equation for this situation and solve it for the initial condition that the pressure is 3000 psi at time t=0 when Phoebe discovers the leak.

I have dp/dt=kp.
dp=kpdt  
Integrating 1/kpdp  
ln absolute value (kp) = t + c
e^ln kp = e^t+c
kp=ce^t
p=c/k e^t
I'm kind of getting lost here. Am I on the right track and how should I go from here?
Thank you.  

Answer
Questioner:   Colleen
Category:  Calculus
 
Subject:  Calculus
Question:  Phoebe is returning to earth in her spaceship when she detects an oxygen tank leak. She knows that the rate of change of the pressure is directly proportional to the pressure of the remaining oxygen.

Use P for pressure and t for time. write a differential equation for this situation and solve it for the initial condition that the pressure is 3000 psi at time t=0 when Phoebe discovers the leak.

I have dp/dt=kp.
dp=kpdt  
Integrating 1/kpdp  
ln absolute value (kp) = t + c
e^ln kp = e^t+c
kp=ce^t
p=c/k e^t
I'm kind of getting lost here. Am I on the right track and how should I go from here?
Thank you.
.....................................
Hi, Colleen,

Yes, I think you did lose your way.  I think it goes like this:

The D.E. should be:

dp
-- = - kp   << Minus , because it is a decrease.
dt  

dp
-- = - k dt
p

ln p = -kt + c

p = e^(-kt + c)

p = e^(-kt) e^c

p = C e^(-kt)   << Big C = e^c

Initial condition:  P = 3000 at  t = 0

3000 = C e^(-k(0))

3000 = C

Equation:  P = 3000 e^(-kt)

That's the best you can do with only one initial condition.  You need a second piece of data to get a second value, such as k.