Question

According to Newton's law of cooling, the temperature u(t) of an object satisfies the differential equation du 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

Answer 1

`u=T+e^(-kt(U_o-T))`

Step-by-step explanation:

Given that

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

Now by separating variables

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

now by taking integration both sides

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

So

`\ln (u-T)=- k\ t +C`

Where C is constant

Given that at t= 0,u=Uo

So

`C=\ln(U_o-T)`

`\ ln(u-T)/(U_o-T)=-k\ t`

`u=T+e^(-kt(U_o-T))`

The temperature at any time t is

`u=T+e^(-kt(U_o-T))`