Applications and Modeling with Differential Equations

Exponential Growth and Decay Model

In the Derivatives and Integrals of Transcendental Functions module exponential growth is defined as a function y = y₀ e^kx where k > 0, whereas exponential decay occurs where k < 0. Both exponential growth and exponential decay can be modeled using differential equations.

y' = ky

Consider the equation y = Ce^kx_. Taking the derivative of this equation produces y'= kCe^kx. Since y = Ce^kx, then y' = ky or equivalently, dy/dx = ky.The next presentation investigates the relationship of the differential equationdy/dx = kyand the exponential function yCe^-kt.

Newton's Law of Cooling states that the rate of change of the temperature of an object at any time t is proportional to the difference in the temperatures of the object and its surroundings at time t. Let T be the temperature at any time t and let T_s be the temperature of the surroundings at time t. Then Newton's Law of Cooling can be expressed as a differential equation:

$LaTeX: \frac{dT}{dt}=k\left(T-T_s\right),\:where\:k\:is\:a\:constant$ equation image indicator

View the presentation below that provides an example of how Newton's Law of Cooling is applied.

Logistic Model

Please take down important notes, such as formulas, and attempt the practice examples on your own before viewing the solutions!

Logistic Differential Equation

The logistic differential equation equation image indicator $LaTeX: \frac{dy}{dt}=ky\left(1-\frac{y}{L}\right)$ , where k and L are positive constants, is used to model populations that grow logistically, i.e., growth that occurs when some upper limit L exists beyond which no additional growth can occur. Notice that if y is small compared to L, then y/L is close to 0 and dy/dt ≈ ky. If y approaches its carrying capacity L, then equation image indicator $LaTeX: \frac{y}{L}\rightarrow1$ and $LaTeX: \frac{dy}{dt}\rightarrow0$ . If the population y lies between 0 and the carrying capacity L, then dy/dt > 0, and the population increases. If y exceeds the carrying capacity L, then dy/dt < 0, and the population decreases.

General and Particular Solutions of Logistic Differential Equations

The logistic differential equation equation image indicator $LaTeX: \frac{dy}{dt}=ky\left(1-\frac{y}{L}\right)$ is separable and a general solution can be found as illustrated below.

$solving a logistic differential equation$

people on stairs talking Consider how a rumor spreads "Tellers" pass it along to "hearers" and hearers become tellers. The rumor spreads slowly at first and then spreads faster when there is an abundance of tellers and hearers. As the hearers become scarce, the rumor slows until it finally stops when everyone knows the rumor.

Applications and Modeling with Differential Equations Practice

Applications and Modeling with Differential Equations: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of applications and modeling with differential equations

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