
3.1 Exponential Growth & Decay; Newton’s Law of Cooling Half Life

3.2 Nonlinear Equations

3.3 Systems of Linear and Nonlinear Differential Equations
Newton’s Law of Cooling
If, at time $t,$ the temperature of an object is $T,$
and the temperature surrounding the object ("ambient temperature")
is $T_m,$
then Newton’s Law of Cooling
is given by the following differential equation:
\[
\frac{dT}{t}=k\left(TT_m\right)
\]
A 1parameter family of solutions for this firstorder, linear equation in $T,$ is
\[
\leftTT_m\right=ce^{kt}
\]
If an initial condition $T\left(0\right)=T_0$ is given, then the solution becomes:
\[
T = T_m  ce^{kt},\ c \gt 0 \qquad \mathrm{provided}\ T_0 \lt T_m \\
T = T_m + ce^{kt}, \ c>0 \qquad \mathrm{provided}\ T_0 \gt T_m
\]
In all cases, $k \lt 0.$
These facts follow from the assumptions, in addition to but
independent of Newton’s Law as stated above, that
a) the object will warm if it’s initial temperature
is less than the ambient temperature, and
b) the object will cool if it’s initial temperature is
greater than the ambient temperature.
HalfLife
The time that it takes for half of the atoms
in a sample of an element to disentegrate into
the atoms of another element (“transmute”).
HalfLife of C14
The half life of C14 is 5600 years.