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

• Jan 25th 2011, 12:39 PM
juanma101285
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.
• Jan 25th 2011, 12:46 PM
pickslides
I get \$\displaystyle y = -Ae^{-x}+1\$

Where \$\displaystyle A=e^c\$