Newton's law of cooling states that d x d t = − k ( x − A ) where x is the temperature, t is time, A is the ambient temperature, and k > 0 is a constant. Suppose that A = A 0 cos ( ω t ) for some constants A 0 and ω . That is, the ambient temperature oscillates (for example night and day temperatures). a) Find the general solution. b) In the long term, will the initial conditions make much of a difference? Why or why not?

(a). The general solution is

(b). The initial condition does not affect the long term.

Explanation:

Given that,

The equation is

Where, x = temperature

t = time

A = ambient temperature  

(a). We need to calculate the general solution

Using given differential equation,

...(I)

Where,

Put the value of A in equation (I)

.....(II)

The integrating factor is given by

Now, multiplying the equation (II) by μ(t) and integrating,

Where, c= constant

....(III)

(b). We need to find the difference in the long term

Using equation (III)

At t = 0,

Now, put the value of c in equation (III)

Now,

For any x(0) ∈ R

So, the initial condition does not affect the long term.

Hence, (a). The general solution is

(b). The initial condition does not affect the long term.