# Thread: ODE Word Problem

1. ## ODE Word Problem

Drug Metabolism The rate at which a drug is absorbed into the bloodstream is modeled by the first-order differential equation:

where $\displaystyle a$ and $\displaystyle b$ are positive constants and $\displaystyle C(t)$ denotes the concentration of the drug in the bloodstream at time . Assume no drug is initially present in the bloodstream.

Find a formula for $\displaystyle C(t)$. You may need to use $\displaystyle a$ and $\displaystyle b$ in your answer.

I think it may be the constants a and b, but something's throwing me off on how to do this. So far I've tried:

$\displaystyle dC/dt = a-bC(t)$
$\displaystyle dC/dt +C(t)= a$

And then I'm guessing I need to use an integrating factor? Or am I just completely wrong here. Any help is greatly appreciated!

2. Originally Posted by cdlegendary
Drug Metabolism The rate at which a drug is absorbed into the bloodstream is modeled by the first-order differential equation:

where $\displaystyle a$ and $\displaystyle b$ are positive constants and $\displaystyle C(t)$ denotes the concentration of the drug in the bloodstream at time . Assume no drug is initially present in the bloodstream.

Find a formula for $\displaystyle C(t)$. You may need to use $\displaystyle a$ and $\displaystyle b$ in your answer.

I think it may be the constants a and b, but something's throwing me off on how to do this. So far I've tried:

$\displaystyle dC/dt = a-bC(t)$
$\displaystyle dC/dt -C(t)= a$

And then I'm guessing I need to use an integrating factor? Or am I just completely wrong here. Any help is greatly appreciated!
$\displaystyle \frac{dC}{dt} = a - b\,C$

$\displaystyle \frac{dC}{dt} + b\,C = a$.

This is first order linear, so use the integrating factor $\displaystyle e^{\int{b\,dt}} = e^{b\,t}$.

Multiplying through gives

$\displaystyle e^{b\,t}\frac{dC}{dt} + b\,e^{b\,t}C = a\,e^{b\,t}$

$\displaystyle \frac{d}{dt}(C\,e^{b\,t}) = a\,e^{b\,t}$

$\displaystyle C\,e^{b\,t} = \int{a\,e^{b\,t}\,dt}$

$\displaystyle C\,e^{b\,t} = \frac{a\,e^{b\,t}}{b} + d$

$\displaystyle C = \frac{a}{b} + d\,e^{-b\,t}$.

When $\displaystyle t = 0, C = 0$.

So $\displaystyle 0 = \frac{a}{b} + d\,e^{0}$

$\displaystyle 0 = \frac{a}{b} + d$

$\displaystyle d = -\frac{a}{b}$.

Therefore

$\displaystyle C = \frac{a}{b} - \frac{a}{b}\,e^{-b\,t}$.

3. How could you find the limiting concentration as t approaches infinity? and at what time does it reach its half life?

I can't do this problem!!! =[