Note for Applied Mathematics - 2 - M-2 By Suhas Mondal

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Ordinary Differential Equation Introduction and Preliminaries There are many branches of science and engineering where differential equations arise naturally. Now 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 dependent variable of (with respect to . The derivatives of The independent variable will be defined for an interval be a ) are denoted by 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 satisfies the differential equation under suitable conditions. 2. Usually (7.1.1) is written as in most of the examples. and the interval which is not mentioned Some examples of differential equations are 1

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1. 2. 3. 4. 5. 6. 7. 8. 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 equation (7.1.1) on if is called a SOLUTION of the differential 1. is differentiable (as many times as the order of the equation) on 2. satisfies the differential equation for all for all If is a solution of an ODE (7.1.1) on Sometimes a solution and . That is, . , we also say that satisfies (7.1.1). is also called an INTEGRAL. EXAMPLE 7.1.5 1. Consider the differential equation , then is differentiable, on . We see that if we take and therefore 2

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Hence, is a solution of the given differential equation for all 2. It can be easily verified that for any constant differential equation on any interval that does not contain the point is a solution of the as the function defined at . Furthere it can be shown that equation whenever the interval contains the point 3. Consider the differential equation that a solution . is not is the only solution for this . on . It can be easily verified of this differential equation satisfies the relation . DEFINITION 7.1.6 (Explicit/Implicit Solution) A solution of the form EXPLICIT SOLUTION (e.g., see Examples 7.1.5.1 and 7.1.5.2). If is called an 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. 3

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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 constant . Remark 7.1.9 The family of functions for an arbitrary 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 arbitrary radius, is an implicit solution. Solution: This family is represented by the implicit relation and (7.1.3) 2. 3. where is a real constant. Then is a solution of the differential equation (7.1.4) 4. 5. The function satisfying (7.1.3) is a one parameter family of solutions or a general solution of (7.1.4). 6. Consider the one parameter family of circles with center at family is represented by the implicit relation and unit radius. The (7.1.5) 7. 8. where is a real constant. Show that satisfies Solution: We note that, differentiation of the given equation, leads to 4