SECTION 10.5: Applications and Modeling

In this section, derivatives related to the original equation with a difference of a constant A are taken into consideration, i.e., dy/dt = k (y-A), where k and A are constants.

Just as the solution to derivatives in the earlier section were found, the solution to the above mentioned derivative can also be determined.

If (y-A) is considered to be an exponential equation, then it must be in the form of C(e^kt).

Earlier we found that the exponential equation dy/dt = ky impliest hat y = C(e^kt). Therefore, this time the solution will vary by a constant A.

dy/dt = k(y-A) implies that y = A + C(e^kt) for any constant C.

Equilibirum solutions:

1. It is constant for all values of the independent variable. The graph is a horizontal line. They can be identified by setting the derivative of the function to zero.

2. It is stable is a small change in the initial conditions gives a solution which tends toward the equillibrium as the independent variable tends to positive infinity.

3. An equilibrium solution is unstable if a small change in the initial conditions gives a solution curve which veers away from the equilibrium as the independent variable tends to positive infinity.

This same principle is applied to Newton's law of heating and cooling by which

Rate of change in temperature = constant * Temperature difference.

This section although not directly applicable to economics is useful in understanding other everyday phenomenon. It is useful in the field of biology in order to know levels of drug concentration in a person. It is useful when solving murder cases or in hospitals when estimating times of death.