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.

adding a(n) to an electric circuit would increase the magnetic field around a wire;

b. solenoid

a solenoid is a type of electromagnet whose purpose is to generate a controlled magnetic field.

if the purpose of the solenoid is instead to impede changes in the electric current, a solenoid can be more specifically classified as an inductor rather than an electromagnet.

the answer is b, have negligible mass