{"id":313,"date":"2018-11-16T06:29:42","date_gmt":"2018-11-16T06:29:42","guid":{"rendered":"http:\/\/msp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=313"},"modified":"2022-01-07T05:40:48","modified_gmt":"2022-01-07T05:40:48","slug":"differential-equations","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp01\/chapter\/differential-equations\/","title":{"rendered":"Differential equations"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\n \r\n\r\nTABLE OF CONTENTS<\/strong>\r\n\r\n \r\n

1. Introduction<\/p>\r\n

2. Classification of differential equations<\/p>\r\n

3. Linear first order differential equations<\/p>\r\n

3.1 Linear homogeneous equations<\/p>\r\n

3.2 Linear inhomogeneous equations<\/p>\r\n

4. Non-linear equations<\/p>\r\n

4.1 Separable equations<\/p>\r\n

4.2 Transformation to separable equations<\/p>\r\n

5. Homogeneous non-linear equations<\/p>\r\n

6. Exact equations<\/p>\r\n

6.1 Integrating factor<\/p>\r\n

6.1.1 Finding integrating factor<\/p>\r\n \r\n\r\n<\/div>\r\n

\r\n\r\nLEARNING OBJECTIVES<\/strong>\r\n

1. The importance of differential equations in engineering, physics and other branches of science, and in social sciences is described.<\/p>\r\n

2. Classification of various types of differential equations is given.<\/p>\r\n

3. Methods of solving linear first order differential equations, both homogeneous and non homogeneous, are described.<\/p>\r\n

4. Nonlinear equations of the separable type are described and methods to convert some equations to separable type are discussed.<\/p>\r\n

5. Next homogeneous nonlinear equations are taken up.<\/p>\r\n

6. Finally the concept of exact differential equations is introduced. Integrating factor that can convert some equations into exact equations is defined and method of finding the integrating factor is given.<\/p>\r\n \r\n\r\n<\/div>\r\n

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D i f f e r e n t i a l\u00a0 E q u a t i o n s<\/strong><\/p>\r\n \r\n\r\n1. Introduction<\/strong><\/span>\r\n\r\n \r\n

Study of differential equations permeates all branches of science. Mathematics in general and differential equations in particular have become more and more relevant in social sciences as well, particularly economics and mathematical ecology. Many physical problems are concerned with relationships between changing quantities. Since rates of change are represented mathematically by derivatives, mathematical models often involve equations relating an unknown function and one or more of its derivatives. Equations involving derivatives of an unknown function are called differential equations<\/em> and our purpose now is to study such equations and their solutions.<\/p>\r\n \r\n

Very often we study situations where certain quantities vary with time. We give a couple of simple examples of time variation as an illustration. One of the simplest and earliest such example is Newton\u2019s law of motion<\/p>\r\n\"\"\r\n

Here we are considering motion in one direction only. F<\/em> is the force acting on a particle of mass m<\/em> and x<\/em> is its position with respect to some reference point. If the position of the particle is known as a function of time, we simply differentiate the function twice and find the force acting on it. However, if the problem is to find the motion of the system for a given force, the problem becomes altogether different. The unknown quantity is now under the differential sign and so this becomes a differential equation.<\/p>\r\n \r\n

When we consider the decay of a radioactive nucleus or the growth of certain species in the wild, the number of members of the population at any given time t is necessarily an integer; models that use differential equations to describe such situations usually rest on the simplifying assumption that the number of members of the population can be regarded as a differentiable function. This is a simple example of mathematical modelling<\/em> of a complex situation to make it tractable.<\/p>\r\n \r\n

In the growth of a species the rate of growth depends both on the birth rate and the death rate of the species. If P<\/em> is the population, in the very simplistic Malthusian model<\/em> the growth rate is taken to be of the simple form<\/p>\r\n\"\"\r\n

To give a more interesting situation, consider a model of two interacting species, a prey and a predator. In the simplest model assume that for the prey the birth rate is proportional to its own numbers and death rate is proportional to the number of predators. For the predator, its growth depends on the number of prey present and decline rate due to death on its own number. If the number of prey is denoted by x<\/em> and that of predator by y<\/em>, then their net rate of growth is determined by the equations<\/p>\r\n\"\"\r\n

This is an example of a system of two differential equations in two unknowns.<\/p>\r\n \r\n\r\n2. Classification of differential equations<\/strong><\/span>\r\n\r\n \r\n

A differential equation is an equation that contains one or more derivatives of an unknown function or functions. A differential equation is an ordinary differential equation<\/em> if it involves an unknown function of only one\u00a0variable, a <\/span>partial differential equation<\/em> if it involves partial derivatives of a function of more than one variable. In these modules we consider only ordinary differential equations; partial differential equations will be taken up later.<\/span><\/p>\r\n

The <\/span>order of a differential equation<\/em> is the order of the highest derivative that it contains. When an equation is polynomial in all the derivatives involved (including the derivative of order zero; that is, the function itself), the power of the highest derivative involved is called the <\/span>degree of a differential equation<\/em>. When in a differential equation, ordinary or partial, the dependent variable and all its derivatives occur only to first degree, the equation is said to be <\/span>linear<\/em>. If there is no term not containing the dependent variable, it is called <\/span>homogeneous<\/em>.<\/span><\/p>\r\n\r\n<\/div>\r\n

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Examples<\/span><\/p>\r\n1. The general ordinary differential equation of order n<\/em> can be written as\r\n\r\n\"\"\r\n\r\n\"\"\r\n

2. As we have seen in an example above,<\/p>\r\n\"\"\r\n

we can have a system of differential equations involving one independent and many dependent variables.<\/p>\r\n3. The differential equation\r\n\r\n\"\"\r\n

is first order, linear, homogeneous, ordinary differential equation, while<\/p>\r\n4.\u00a0\"\"\r\n

is first order, linear, inhomogeneous, ordinary differential equation.<\/p>\r\n5. The equation\r\n\r\n\"\"\r\n

is second order, linear, inhomogeneous, ordinary differential equation.<\/p>\r\n6. The equation\r\n\r\n\"\"\r\n\r\n\"\"\r\n

7. The equation<\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

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8. As examples of linear first and second order partial differential equations in two or three independent variables, we have the divergence or the Laplace equations:<\/p>\r\n\"\"\r\n

Given any differential equation, ordinary or partial, linear or non-linear etc., by finding a solution of the equation we mean finding a function y<\/em>(x<\/em>), which when substituted in the given equation satisfies in identically.<\/p>\r\n\r\n