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

1. ## 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.

2. I get $\displaystyle y = -Ae^{-x}+1$

Where $\displaystyle A=e^c$