6 Differential equations

TABLE OF CONTENTS

1. Introduction

2. Classification of differential equations

3. Linear first order differential equations

3.1 Linear homogeneous equations

3.2 Linear inhomogeneous equations

4. Non-linear equations

4.1 Separable equations

4.2 Transformation to separable equations

5. Homogeneous non-linear equations

6. Exact equations

6.1 Integrating factor

6.1.1 Finding integrating factor

LEARNING OBJECTIVES

1. The importance of differential equations in engineering, physics and other branches of science, and in social sciences is described.

2. Classification of various types of differential equations is given.

3. Methods of solving linear first order differential equations, both homogeneous and non homogeneous, are described.

4. Nonlinear equations of the separable type are described and methods to convert some equations to separable type are discussed.

5. Next homogeneous nonlinear equations are taken up.

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.

D i f f e r e n t i a l  E q u a t i o n s

1. Introduction

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 and our purpose now is to study such equations and their solutions.

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’s law of motion

Here we are considering motion in one direction only. F is the force acting on a particle of mass m and x 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.

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 of a complex situation to make it tractable.

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 is the population, in the very simplistic Malthusian model the growth rate is taken to be of the simple form

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 and that of predator by y, then their net rate of growth is determined by the equations

This is an example of a system of two differential equations in two unknowns.

2. Classification of differential equations

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 if it involves an unknown function of only one variable, a partial differential equation 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.

The order of a differential equation 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 degree of a differential equation. 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 linear. If there is no term not containing the dependent variable, it is called homogeneous.

Examples

1. The general ordinary differential equation of order n can be written as

2. As we have seen in an example above,

we can have a system of differential equations involving one independent and many dependent variables.

3. The differential equation

is first order, linear, homogeneous, ordinary differential equation, while

4. 

is first order, linear, inhomogeneous, ordinary differential equation.

5. The equation

is second order, linear, inhomogeneous, ordinary differential equation.

6. The equation

7. The equation

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:

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(x), which when substituted in the given equation satisfies in identically.

• We make a general declaration here that throughout this module we assume, without stating it explicitly every time, that the functions involved and the solution etc., are continuous and differentiable functions.

3.  Linear first order differential equations

In the rest of this module we take up first order differential equations. The general form of the first order equation is (prime refers to derivative with respect to x)

However, we will first take up the special case of homogeneous and inhomogeneous linear equations. The theory of linear equations is the most highly developed; many properties of the solutions have been investigated and general methods of finding solutions exist.

The most general linear first order equation has the form

If further q(x) = 0, the equation is homogeneous.  The homogeneous equation definitely has a solution:  y(x) ≡ 0. This is the trivial solution. Any other solution is nontrivial.

If p(x) = 0, the equation takes the form

where c is some arbitrary constant. Whenever the solution can be written as an integral, we say the equation has been solved in principle. The solution in this case is a family of functions since c is an arbitrary constant.

3.1 Linear homogeneous equations

Let us look first at the homogeneous equation:

   

 This is the most general solution of the linear, homogeneous, first order differential equation. The trivial solution corresponds to K = 0. This is basically the method of separation of variables, since terms depending on y have been brought to one side and those depending on x alone to the other.

Example: Newton’s law of cooling

3.2 Linear inhomogeneous equations

Now let us consider the linear inhomogeneous equation (2). The corresponding or associated homogeneous equation

Substitute it back into (1), and we have

Examples

4. Non-linear equations

Any equation that is not linear is non-linear. The general problem of non-linear equations is rather complicated. There are no general methods that can tackle all non-linear equations. The solution to an initial value problem may or may not exist and when it does, may or may not be unique. However there are certain class of equations for which general methods do exist. One such class is that of separable equations.

4.1 Separable equations

A first order equation is said to be separable if it can be put in the form:

This is the required implicit solution.  The equation cannot be solved for y in terms of x.  Solution to initial value problem is

4.2 Transformation to separable equations

Certain equations are not separable as they stand but can be converted into that form by method of variation of parameters similar to that employed for the linear inhomogeneous equations. Let us consider the example of Bernoulli equation:

5. Homogeneous non-linear equations

The general first order differential equation (1) is said to be homogeneous if F(x,y) can be written as a function of (y/x) alone. Then the differential equation becomes

For example the equation

is homogeneous.

• This definition of homogeneity is very different from the usual notion which we employed in the definition of linear homogeneous equations. These two different notions of homogeneity are historical; however, but for this section we will not need this definition of homogeneity.

To solve equation (17), we put

Example

This is the required general solution.

6. Exact equations

An ordinary differential equation of the first order and first degree may be expressed in the form of a total differential:

Proof

If the expression on the left side of equation (20) is an exact differential du, then

Examples

except along the real axis.  Thus there is no region in the plane in which the condition is satisfied.

6.1 Integrating factor

Sometimes the given equation is not exact but can be made so by multiplying by a factor u(x, y), called integrating factor which makes the equation exact. That is

becomes a perfect differential.

• It is possible that a solution of the equation with integrating factor is not a solution of the original equation, or a solution exists even when the integrating factor does not. However we will ignore such niceties here.

6.1.1 Finding integrating factor

If u is to be integrating factor, then from the condition of exactness, we have

Here the subscript refers to partial differentiation. In general this equation is of little value because to find u we need to solve a partial differential equation, an infinitely more arduous task. However we can try to find an integrating factor which is a product of a function of x and of y if such an integrating factor exists. So let

If we substitute equation (26) into (25), the condition becomes

The situation is even now rather complicated. There exist simple criteria for the existence of the integrating factor provided we assume it to be a function of only one of the variables. We state the following theorem without proof:

Theorem

Example

is independent of x.  Hence

To find the primitive

SUMMARY

• With this module we begin our study of ordinary differential equations. We emphasize the importance of differential equations in engineering, physics and other branches of science and social sciences and illustrate with a couple of simple examples.
• Next we classify differential equations into various types; first and higher order equations, linear and nonlinear equations, ordinary and partial differential equations and system of equations etc. We give illustrative examples of each type.
• In the rest of this module we study first order equations only. We first describe methods of solving linear first order differential equations, both homogeneous and non homogeneous and provide a few examples.
• Next nonlinear equations of the separable type are taken up. We describe a method to convert some equations to separable type.
• After that we define what is meant by homogeneous nonlinear equations and give a method to solve such equations.
• Finally we introduce the concept of exact differential equations. Some equations that are not exact can be converted into that form on multiplying by an integrating factor. We discuss a method of finding the integrating factor in such cases.