Let the temperature of some body at time  t  be  T(t), and let the ambient temperature have a time independent value  A.

1. Assume that  T(0) = T0 > A.
   Then the cooling of a body can be described by Newton's Model of Cooling, which states that the time rate of change of the temperature is proportional to the difference between temperatures:
   .
   The value of the proportionality constant  k  is determined by the properties of the object and ambient. However, in any circumstances    > 0.

   Separation of variables and integration gives the general integral of the differential equation:

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   Since  T(0) = T0  then
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   and so
   .
   Therefore,
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   which yields the particular solution:

   .

   One can see that  T(t) approaches the limit  A  as   .
2. Let us assume now that  T0 < A.
   Then the heating of a body is described by the following initial value problem:

   .

   In a similar way as above we obtain the particular solution, which obeys the initial condition:

   .

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Indefinite Integrals Definite Integrals Differential Equations