Couple differential equations

• February 1st 2011, 02:14 PM
Jeonsah
Couple differential equations
Hey guys, Ive been doing some Diffy Q work and I have a few problems I am stuck with. I figured id post up some of the ones Im stuck on and hopefully someone can help. Im having a hard time understanding what is going on with these problems and what they actually mean in words.

1.
$\frac{dP}{dt} = P(1-P); P = \frac{Ce^t}{1 + Ce^t}$

In the problem above a one-paremeter family of solutions of the DE $P' = P(1 - P)$ is given. Does any solution curve pass through the point (0,3)? Through the point (0,1)?

2. Newton's Law of Cooling/Warming

A cup of coffee cools according to Newton's Law of cooling
$\frac{dT}{dt} = K(T-Tm)$
Use date from the graph of the temperature T(t) to estimate the constants Tm, and T0(T naught), and K in a model of the form of a first-order initial-value problem:
$\frac{dT}{dt} = K(T-Tm), T(0) = T0$

http://i909.photobucket.com/albums/a...andWarming.jpg

Thank you for you help. Im just lost at the terminology of differential equations and what exactly what im trying to be able to understand. Any help is appreciated.
• February 1st 2011, 02:22 PM
dwsmith
For 1, solve P(t) for (0,3) and (0,1).

For 2, T_{m} looks to equal 75, and T_{0} looks to equal 175.
• February 1st 2011, 02:26 PM
Jeonsah
what does it mean by " one parameter family of solutions?" and how I go about solving for (0,3) and (0,1)?
Im sorry if I sound dumb but this stuff is very hard to read/understand for me. I am use to just solving a bunch of calculus problems without thinking too much.
• February 1st 2011, 02:39 PM
dwsmith
Quote:

Originally Posted by Jeonsah
what does it mean by " one parameter family of solutions?" and how I go about solving for (0,3) and (0,1)?
Im sorry if I sound dumb but this stuff is very hard to read/understand for me. I am use to just solving a bunch of calculus problems without thinking too much.

$P(t) = \frac{Ce^t}{1 + Ce^t}$

$P(0) = \frac{Ce^0}{1 + Ce^0}=3$

Solve for c and you will have a solution that passes through (0,3). Do the same thing for (0,1).

$\displaystyle \frac{dT}{dt} = K(T-75), \ T(0) = 175$

Estimate K by solving the DE and solving for K.