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 $x$ be an independent variable and let $y$ be a dependent variable of $x$ . The derivatives of $y$ (with respect to $x$ ) are denoted by

$$y^{\prime}= \frac{dy}{dx}, \;y^{\prime\prime}= \frac{d^2y}{dx^2}, \ldots, y^{(k)}= \frac{d^{(k)}y}{d x^{(k)}} \;\; {\mbox{ for }} \;\; k \geq 3.$$

The independent variable will be defined for an interval $I;$ where $I$ is either ${\mathbb{R}}$ or an interval $a < x < b \subset {\mathbb{R}}.$ 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

$$f \bigl(x, y, y^\prime, \ldots, y^{(n)}\bigr) = 0 \hspace{1in} {\mbox{ for }} \;\; x \in I$$ (7.1.1)

is called an ORDINARY DIFFERENTIAL EQUATION; where $f$ is a known function from $I \times {\mathbb{R}}^{n+1}$ to ${\mathbb{R}}.$

Remark 7.1.2  

1. The aim of studying the ODE (7.1.1) is to determine the unknown function $y$ which satisfies the differential equation under suitable conditions.
2. Usually (7.1.1) is written as $f \bigl(x, y, y^\prime, \ldots, y^{(n)}\bigr) = 0,$ and the interval $I$ is not mentioned in most of the examples.

Some examples of differential equations are

1. $y^{\prime} = 6 \sin x + 9;$
2. $y^{\prime\prime} + 2 y^2 = 0;$
3. $\sqrt{y^{\prime}} = \sqrt{x} + \cos y;$
4. ${(y^\prime)}^2 + y = 0.$
5. $y^\prime + y = 0.$
6. $y^{\prime\prime} + y = 0.$
7. $y^{(3)} = 0.$
8. $y^{\prime\prime} + m \sin \bigl(y\bigr) = 0.$

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 $y = \phi(x)$ is called a SOLUTION of the differential equation (7.1.1) on $I$ if

1. $\phi(x)$ is differentiable (as many times as the order of the equation) on $I$ and
2. $\phi(x)$ satisfies the differential equation for all $x \in I$ . That is, $f \bigl(x, \phi(x), \phi^\prime(x), \ldots, \phi^{(n)}(x)\bigr) = 0$ for all $x \in I$ .

If $y = \phi(x)$ is a solution of an ODE (7.1.1) on $I$ , we also say that $\phi(x)$ satisfies (7.1.1). Sometimes a solution $y = \phi(x)$ is also called an INTEGRAL.

EXAMPLE 7.1.5  

1. Consider the differential equation $y^{\prime} + 2 y = 0$ on ${\mathbb{R}}$ . We see that if we take $y(x) = c e^{-2x}$ , then $y(x)$ is differentiable, $y^{\prime}(x) = - 2c e^{-2x}$ and therefore
   $$y^{\prime}(x) + 2 y(x) = - 2c e^{-2x} + 2c e^{-2x} = 0 \;\; {\mbox{ for all }} \; x \in {\mathbb{R}}.$$
   Hence, $y(x) = c e^{-2x}$ is a solution of the given differential equation for all $x \in {\mathbb{R}}$ .
2. It can be easily verified that for any constant $a \in {\mathbb{R}}, \;\; y = \displaystyle\frac{a}{1 - x}$ is a solution of the differential equation
   $$(1-x) y^{\prime} - y = 0$$
   on any interval that does not contain the point $x = 1$ as the function $y = \displaystyle\frac{a}{1 - x}$ is not defined at $x = 1$ . Furthere it can be shown that $y(x) \equiv 0$ is the only solution for this equation whenever the interval $I$ contains the point $x = 1$ .
3. Consider the differential equation $(x-1) + y y^\prime = 0$ on $[-1, 1]$ . It can be easily verified that a solution $y = \phi(x)$ of this differential equation satisfies the relation $(x-1)^2 + \phi^2(x) = 1$ .

DEFINITION 7.1.6 (Explicit/Implicit Solution)   A solution of the form $y = \phi(x)$ is called an EXPLICIT SOLUTION (e.g., see Examples 7.1.5.1 and 7.1.5.2). If $y$ is given by an implicit relation $h(x, y) = 0$ and satisfies the differential equation, then $y$ 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 $n,$ we expect $n ( n \geq 1)$ arbitrary constants.

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

$$f(x, y, y^\prime) = 0$$ (7.1.2)

and move to higher orders later.

DEFINITION 7.1.8 (General Solution)   A function $\phi(x,c)$ is called a general solution of (7.1.2) on an interval $I \subset {\mathbb{R}},$ if $\phi(x,c)$ is a solution of (7.1.2) for each $x \in I,$ for an arbitrary constant $c$ .

Remark 7.1.9   The family of functions

$$\{\phi(\cdot,c) : \; c {\mbox{ is a constant such that }} \phi(\cdot,c) {\mbox{ is well defined}} \}$$

is called a one parameter family of functions and $c$ 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 $(1,0)$ and arbitrary radius, $a$ is an implicit solution.
   Solution: This family is represented by the implicit relation

   $$(x-1)^2 + y^2 = a^2,$$ (7.1.3)

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

   $$(x-1) + y \frac{dy}{dx} = 0.$$ (7.1.4)

   
   
   The function $y$ 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 $(c,0)$ and unit radius. The family is represented by the implicit relation

   $$(x-c)^2 + y^2 = 1,$$ (7.1.5)

   
   
   where $c$ is a real constant. Show that $y$ satisfies $\bigl( y y^{\prime} \bigr)^2 + y^2 = 1.$
   Solution: We note that, differentiation of the given equation, leads to
   $$(x-c) + y y^{\prime} = 0.$$
   Now, eliminating $c$ from the two equations, we get
   $$(y y^{\prime})^2 + y^2 = 1.$$

In Example 7.1.10.1, we see that $y$ is not defined explicitly as a function of $x$ but implicitly defined by (7.1.3). On the other hand $y = \displaystyle \frac{1}{1-x}$ 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 $x, \; y$ and the slope of the function $y$ at the point $x.$ For instance, let us find the equation of the curve passing through $(0, \displaystyle\frac{1}{2})$ and whose slope at each point $(x,y)$ is $- \displaystyle\frac{x}{4y}.$ If $y$ is the required curve, then $y$ satisfies

$$\frac{dy}{dx} = - \frac{x}{4y}, \; y(0) = \frac{1}{2}.$$

It is easy to verify that $y$ satisfies the equation $x^2 + 4 y^2 = 1.$

EXERCISE 7.1.11  

1. Find the order of the following differential equations:
   1. $y^2 + \sin (y^{\prime}) = 1.$
   2. $y + (y^{\prime})^2 = 2 x.$
   3. $(y^{\prime})^3 + y^{\prime\prime} - 2 y^4 = -1.$
2. Show that for each $k \in {\mathbb{R}}, \; y = k e^x$ is a solution of $y^\prime =y.$
3. Find a differential equation satisfied by the given family of curves:
   1. $y = m x,\; m$ real (family of lines).
   2. $y^2 = 4 a x, \; a$ real (family of parabolas).
   3. $x = r^2 \cos \theta, \; y = r^2 \sin \theta, \; \theta$ is a parameter of the curve and $r$ is a real number (family of circles in parametric representation).
4. Find the equation of the curve $C$ which passes through $(1,0)$ and whose slope at each point $(x,y)$ is $$\frac{-x}{y}.$$