Introduction and Preliminaries

There are many branches of science and engineering where differential equations arise naturally. Now a days, it finds applications in many areas including medicine, economics and social sciences. In this context, the study of differential equations assumes importance. In addition, in the study of differential equations, we also see the applications of many tools from analysis and linear algebra. Without spending more time on motivation, (which will be clear as we go along) let us start with the following notations. Let be an independent variable and let be a dependent variable of . The derivatives of (with respect to ) are denoted by

The independent variable will be defined for an interval where is either or an interval With these notations, we are ready to define the term differential equation".

A differential equation is a relationship between the independent variable and the unknown dependent function along with its derivatives. More precisely, we have the following definition.

DEFINITION 7.1.1 (Ordinary Differential Equation, ODE)   An equation of the form

 (7.1.1)

is called an ORDINARY DIFFERENTIAL EQUATION; where is a known function from to

Remark 7.1.2
1. The aim of studying the ODE (7.1.1) is to determine the unknown function which satisfies the differential equation under suitable conditions.
2. Usually (7.1.1) is written as and the interval is not mentioned in most of the examples.

Some examples of differential equations are

DEFINITION 7.1.3 (Order of a Differential Equation)   The ORDER of a differential equation is the order of the highest derivative occurring in the equation.

In Example 7.1, the order of Equations 1, 3, 4, 5 are one, that of Equations 2, 6 and 8 are two and the Equation 7 has order three.

DEFINITION 7.1.4 (Solution)   A function is called a SOLUTION of the differential equation (7.1.1) on if
1. is differentiable (as many times as the order of the equation) on and
2. satisfies the differential equation for all . That is, for all .

If is a solution of an ODE (7.1.1) on , we also say that satisfies (7.1.1). Sometimes a solution is also called an INTEGRAL.

EXAMPLE 7.1.5
1. Consider the differential equation on . We see that if we take , then is differentiable, and therefore

Hence, is a solution of the given differential equation for all .
2. It can be easily verified that for any constant is a solution of the differential equation

on any interval that does not contain the point as the function is not defined at . Furthere it can be shown that is the only solution for this equation whenever the interval contains the point .
3. Consider the differential equation on . It can be easily verified that a solution of this differential equation satisfies the relation .

DEFINITION 7.1.6 (Explicit/Implicit Solution)   A solution of the form is called an EXPLICIT SOLUTION (e.g., see Examples 7.1.5.1 and 7.1.5.2). If is given by an implicit relation and satisfies the differential equation, then is called an IMPLICIT SOLUTION (e.g., see Example 7.1.5.3).

Remark 7.1.7   Since the solution is obtained by integration, we may expect a constant of integration (for each integration) to appear in a solution of a differential equation. If the order of the ODE is we expect arbitrary constants.

To start with, let us try to understand the structure of a first order differential equation of the form

 (7.1.2)

and move to higher orders later.

DEFINITION 7.1.8 (General Solution)   A function is called a general solution of (7.1.2) on an interval if is a solution of (7.1.2) for each for an arbitrary constant .

Remark 7.1.9   The family of functions

is called a one parameter family of functions and is called a parameter. In other words, a general solution of (7.1.2) is nothing but a one parameter family of solutions of (7.1.2).

EXAMPLE 7.1.10
1. Determine a differential equation for which a family of circles with center at and arbitrary radius, is an implicit solution.
Solution: This family is represented by the implicit relation

 (7.1.3)

where is a real constant. Then is a solution of the differential equation

 (7.1.4)

The function satisfying (7.1.3) is a one parameter family of solutions or a general solution of (7.1.4).
2. Consider the one parameter family of circles with center at and unit radius. The family is represented by the implicit relation

 (7.1.5)

where is a real constant. Show that satisfies
Solution: We note that, differentiation of the given equation, leads to

Now, eliminating from the two equations, we get

In Example 7.1.10.1, we see that is not defined explicitly as a function of but implicitly defined by (7.1.3). On the other hand is an explicit solution in Example 7.1.5.2.

Let us now look at some geometrical interpretations of the differential Equation (7.1.2). The Equation (7.1.2) is a relation between and the slope of the function at the point For instance, let us find the equation of the curve passing through and whose slope at each point is If is the required curve, then satisfies

It is easy to verify that satisfies the equation

EXERCISE 7.1.11
1. Find the order of the following differential equations:
2. Show that for each is a solution of
3. Find a differential equation satisfied by the given family of curves:
1. real (family of lines).
2. real (family of parabolas).
3. is a parameter of the curve and is a real number (family of circles in parametric representation).
4. Find the equation of the curve which passes through and whose slope at each point is

A K Lal 2007-09-12