section_1_1.pdf - Elementary Differential Equations with...

This preview shows page 1 - 13 out of 64 pages.

Elementary Differential Equations with Boundary Value Problems, 6th ed. Section 1.1 Differential Equations and Mathematical Models C. Henry Edwards David E. Penney
Objectives Introduction Differential Equations Example Goals of the Study of Differential Equations Example
Objectives Differential Equations and Mathematical Models Introduction Examples Infinite Family of Solutions Example Initial Conditions Mathematical Models A Three-Step Cycle Mathematical Models
Objectives (cont’d) Examples and Terminology Examples Existence and Uniqueness Terminology Examples Ordinary and Partial Differential Equations Initial Value Problems Example Finding Solutions
Introduction
Differential Equations Changing Quantities The laws of the universe are written in the language of mathematics. Algebra is sufficient to solve many static problems, but the most interesting natural phenomena involve change and are described by equations that relate changing quantities.
Differential Equations Derivative as Rate of Change Because the derivative dx / dt = f 0 ( t ) of the function f is the rate at which the quantity x = f ( t ) is changing with respect to the independent variable t . . . . . . it is natural that equations involving derivatives are frequently used to describe the changing universe. An equation relating an unknown function and one or more of its derivatives is called a differential equation .
Example Example 1 The equation dx dt = x 2 + t 2 involves the unknown function x ( t ) and its first derivative x 0 ( t ) . The equation d 2 y dx 2 + 3 dy dx + 7 y = 0 involves the unknown function y ( x ) and its first two derivatives.
Goals of the Study of Differential Equations Three Goals The study of differential equations has three principal goals: 1. To discover the differential equation that describes a specified physical situation. 2. To find—either exactly or approximately—the appropriate solution of that equation. 3. To interpret the solution that is found.
Goals of the Study of Differential Equations Unknowns In algebra, we typically seek the unknown numbers that satisfy an equation such as x 3 + 7 x 2 - 11 x + 41 = 0 . By contrast, in solving a differential equation, we are challenged to find the unknown functions y = y ( x ) for which an identity such as y 0 ( x ) = 2 xy ( x ) —that is, the differential equation dy dx = 2 xy —holds on some interval of real numbers. Ordinarily, we will want to find all solutions of the differential equation, if possible.
Example Example 2 If C is a constant and y ( x ) = Ce x 2 , then dy dx = C 2 xe x 2 = ( 2 x ) Ce x 2 = 2 xy . Thus every function y ( x ) of the above form satisfies —and thus is a solution of—the differential equation dy dx = 2 xy for all x .
Example Example 2 (cont’d) In particular, the equation y ( x ) = Ce x 2 defines an infinite family of different solutions of this differential equation, one for each choice of the arbitrary constant C .

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture