We define ordinary differential equations and what it means for a function to be a solution to such an equation.
Basic Concepts
What is a Differential Equation?
A differential equation is an equation that contains one or more derivatives of an unknown function. The order of a differential equation is the order of the highest derivative that it contains. A differential equation is an ordinary differential equation if it involves an unknown function of only one variable, or a partial differential equation if it involves partial derivatives of a function of more than one variable. For now we’ll consider only ordinary differential equations, and we’ll just call them differential equations.
Throughout this text, all variables and constants are real numbers unless it’s stated otherwise. We’ll usually use for the independent variable unless the independent variable is time; then we’ll use .
The simplest differential equations are first order equations of the form where is a known function of . We already know from calculus how to find functions that satisfy this kind of equation. For example, if then where is an arbitrary constant. If we can find functions that satisfy equations of the form
by repeated integration. Again, this is a calculus problem.Except for illustrative purposes in this section, there’s no need to consider differential equations like (eq:1.2.1). We’ll usually consider differential equations that can be written as
where at least one of the functions , , …, actually appears on the right. Here are some examples: Although none of these equations is written as in (eq:1.2.2), all of them can be written in this form:Solutions of Differential Equations
A solution of a differential equation is a function that satisfies the differential equation on some open interval; thus, is a solution of (eq:1.2.2) if is times differentiable and for all in some open interval . In this case, we also say that is a solution of on . Functions that satisfy a differential equation at isolated points are not interesting. For example, satisfies if and only if or , but it’s not a solution of this differential equation because it does not satisfy the equation on an open interval.
The graph of a solution of a differential equation is a solution curve. More generally, a curve is said to be an integral curve of a differential equation if every function whose graph is a segment of is a solution of the differential equation. Thus, any solution curve of a differential equation is an integral curve, but an integral curve need not be a solution curve.
The graph of (eq:1.2.5) appears below
The part of the graph of (eq:1.2.5) on is a solution curve of (eq:1.2.6), as is the part of the graph on .
Since the constants in (eq:1.2.10) are arbitrary, so are the constants Therefore Example example:1.2.4 actually shows that all solutions of (eq:1.2.9) can be written as where we renamed the arbitrary constants in (eq:1.2.10) to obtain a simpler formula. As a general rule, arbitrary constants appearing in solutions of differential equations should be simplified if possible. You’ll see examples of this throughout the text.
Initial Value Problems
In Example example:1.2.4 we saw that the differential equation has an infinite family of solutions that depend upon the arbitrary constants . In the absence of additional conditions, there’s no reason to prefer one solution of a differential equation over another. However, we’ll often be interested in finding a solution of a differential equation that satisfies one or more specific conditions. The next example illustrates this.
We can rewrite the problem considered in Example example:1.2.5 more briefly as
We call this an initial value problem. The requirement is an initial condition. Initial value problems can also be posed for higher order differential equations. For example,
is an initial value problem for a second order differential equation where and are required to have specified values at . In general, an initial value problem for an -th order differential equation requires and its first derivatives to have specified values at some point . These requirements are the initial conditions.We’ll denote an initial value problem for a differential equation by writing the initial conditions after the equation, as in (eq:1.2.11). For example, we would write an initial value problem for (eq:1.2.2) as
Consistent with our earlier definition of a solution of the differential equation in (eq:1.2.12), we say that is a solution of the initial value problem (eq:1.2.12) if is times differentiable and for all in some open interval that contains , and satisfies the initial conditions in (eq:1.2.12). The largest open interval that contains on which is defined and satisfies the differential equation is the interval of validity of .
Similarly, the interval of validity of (eq:1.2.14) as a solution of (eq:1.2.16) is , since this is the largest interval that contains on which (eq:1.2.14) is defined.
Free Fall Under Constant Gravity
The term initial value problem originated in problems of motion where the independent variable is (representing elapsed time), and the initial conditions are the position and velocity of an object at the initial (starting) time of an experiment.
- (a)
- Construct a mathematical model for the motion of the object in the form of an initial value problem for a second order differential equation, assuming that the altitude and velocity of the object at time are known. Assume that gravity is the only force acting on the object.
- (b)
- Solve the initial value problem derived in Part part:ex1.2.8partA to obtain the altitude as a function of time.
Part part:ex1.2.8partB. Integrating (eq:1.2.17) twice yields
Text Source
Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)