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
where T(t) is the temperature at time t, k is a positive constant, and 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
Thereby, show that the differential equation DT/dt may be scaled to the differential equation
(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.
, given that . Is this correct.