Question

According to Newton’s law of cooling, the temperature u(t) of an object satisfies the differential equationdu/dt= −k(u−T)where T is the constant ambient temperature and k is a positive constant. Suppose that the initial temperature of the object is u(0)= u0.(a) Find the temperature of the object at any time. (I know how to resolve this)

Answer 1

`u(t)=T+(u_(0)-T})e^(-kt)`

Explanation:

We know:

`(du)/(dt) = -k(u-T)`

We integrate in order to find u(t):

`\int\limits^u_{u_(0)} {(1)/(-k(u-T)) \, du } = \int\limits^t_0 \, dt`

`ln((u-T)/(u_(0)-T) )=-kt\\`

`u(t)=T+(u_(0)-T})e^(-kt)`