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Math Help - Couple differential equations

  1. #1
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    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.
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  2. #2
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    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.
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  3. #3
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    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.
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  4. #4
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    Quote Originally Posted by Jeonsah View Post
    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.
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