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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

Respuesta :

Answer:

[tex]u=T+e^{-kt(U_o-T)}[/tex]

Explanation:

Given that

[tex]\dfrac{du}{dt}=-k(u-T)[/tex]

Now by separating variables

[tex]\dfrac{du}{(u-T)}=-k\ dt[/tex]

now by taking integration both sides

[tex]\int \dfrac{du}{(u-T)}=-\int k\ dt[/tex]

So

[tex]\ln (u-T)=- k\ t +C[/tex]

Where C is constant

Given that at t= 0,u=Uo

So

[tex]C=\ln(U_o-T)[/tex]

[tex]\ ln\dfrac{u-T}{U_o-T}=-k\ t[/tex]

[tex]u=T+e^{-kt(U_o-T)}[/tex]

The temperature at any time t is

[tex]u=T+e^{-kt(U_o-T)}[/tex]

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