Differential Equations and Their Application Module Overview

Introduction

Differential equations provide the impetus for some of the most important and useful applications of calculus. They arise in the process of modeling some phenomenon related to change, motion, or growth. Although it is not always possible to find an explicit algebraic solution of a differential equation through analytic means, technology paves the way for graphical and numerical approaches to a solution.

Essential Questions

What is a differential equation?
How are differential equations used to model real-world problems?
What is a slope field and how can it be used to find solutions to differential equations?
How do initial conditions affect the solution to a differential equation?
How is graphing technology used to investigate the effects of different initial conditions on the solution to a differential equation?
What is a homogeneous differential equation and how can it be transformed into a separable differential equation?
What is a logistic differential equation and when is it used in modeling?

Key Terms

The following key terms will help you understand the content in this module.

Carrying capacity - The maximum value of a population that an environment can support as time increases.

Differential equation - An equation that contains an unknown function and one or more of its derivatives.

General solution - A family of solutions of a differential equation.

Homogeneous differential equation - An equation of the form M(x,y)dx + N(x,y)dy = 0 where M and N are homogenous functions of the same degree.

Homogeneous function of degree n - A function given by f(x,y) where f(tx,ty)=tⁿf(x,y) with n a real number

Logistic differential equation - An equation of the form $LaTeX: \frac{dy}{dt}=ky\left(1=\frac{y}{L}\right)$ , where k and L are positive constants, that models growth of a population approaching an upper limit (L is the carrying capacity).

Order - The ordinal number of the highest derivative that occurs in a differential equation.

Particular solution - The one solution of a differential equation obtained from initial conditions.

Separable equation - A first-order differential equation in which the expression for dy/dx can be factored as a function of x times a function of y, dy/dx = g(x) f(y).

Separation of variables - A technique for solving a differential equation that involves rewriting the equation so that each variable appears on only one side of the equation.

Slope field - A graph of short tangent line segments centered at each of many grid points (x,y) that approximates a family of solution curves to y' = f(x, y).

Solution - A differentiable function f defined on an interval I of x values such that when y = f(x) and its derivative(s) are substituted into a differential equation, the resulting equation is true for all x over the interval I.

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