• 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: \[ \frac{dT}{t}=k\left(T-T_m\right) \] A 1-parameter family of solutions for this first-order, linear equation in $T,$ is \[ \left|T-T_m\right|=ce^{kt} \] 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.


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”).

Half-Life of C-14

The half life of C-14 is 5600 years.