1. ## 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:
$\frac{dT}{dt} = K(T-Tm), T(0) = T0$

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.

2. 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.

3. 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.

4. 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.