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Thread: Problem with a differential equation (using Newton's Law of Cooling)

  1. #1
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    Problem with a differential equation (using Newton's Law of Cooling)

    Hi, I have the following problem. I have done question D, but I don't know how to do the other 3 :-/. Any help would be very much appreciated, thanks. I know I am asking for a lot, but I'll be happy with whatever help I can get...

    The equation for Newton's Law of Cooling is

    $\displaystyle dT/dt = -k*[T(t)-T_0]$

    where T(t) is the temperature at time t, k is a positive constant, and $\displaystyle T_0$ is the (constant) temperature of the environment.

    (a) INtroduce the new variables y=aT and x=bt where a and b are constants, and show that

    $\displaystyle (dy)/(dx) = (a/b)*(dT)/(dt)$

    Thereby, show that the differential equation DT/dt may be scaled to the differential equation

    $\displaystyle dy/dx=-y+1$ (let's call it equation 1)

    by choosing appropriate values for a and b.

    (b) Sketch the direction field of equation 1.
    I thought I had to solve question d first and then use the result, but that won't help me in any way because of the constant C.

    (c) Find the equilibrium solution of equation 1 (the solution where y is constant?)

    (d) Obtain the general solution of equation 1.

    I got:

    $\displaystyle y=1-e^{(-x)}*e^{(-C')}=1-e^{(-x)}*C$, given that $\displaystyle C=e^{(-C')}$. Is this correct.
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  2. #2
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    I get $\displaystyle y = -Ae^{-x}+1$

    Where $\displaystyle A=e^c$
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