Chapter 1

Ordinary Differential Equation:

Introduction

by

Dwijendra Narain Pandey

Department of Mathematics
Indian Institute of Technology Roorkee
Roorkee 247667, Uttarakhand.
E-mail: dwij.iitk@gmail.com
Module 1: Introduction.

1 Introduction
Differential equations are used to express many general law of nature and have
many applications in physical, biological, social, economical and other dynami-
cal systems. In particular, the origin of differential equations may be considered
as the efforts of Newton to illustrate the motion of particles.
These equations may provide many useful information about the system if
the equation were formed incorporating the various important factors of the
system. Let y = f (t) be a given function of independent variable t then the
change in y with respect to t can be represented as the derivative dy dt . In many
real world process, the variables representing the important factors of the pro-
cess and their rate of change are related to each other in terms of the basic
scientific principles of the process. The representation of the relation in terms
of mathematical forms often turns out be a differential equation and this may
explain the vital utility of the differential equation in most of the dynamical
systems of modern science and technology.
The following very simple example of simple pendulum may illustrates the
importance of the above remarks. Though the example may look simple but it
is prototype of many other complicated physical systems. Let us recall some im-
portant points of the Newtonian models for the motion of a system of particles.
Here we assume that a body(particle) is represented as a point having mass.
According to the Newton’s first law of motion: In the absence of forces, the
motion of each particle is un-accelerated. Therefore the presence of acceleration
in motion of a particle is the symbol of presence of force acting on the particle.
So, according to Newton’s second law of motion: If F is the force acting on a
d
particle of mass m moving with a velocity v then F = dt (mv) = ma, where a
is the acceleration of the particle.
Example 1. Simple Pendulum: A weight of mass m is attached to a very
light and rigid rod of length l to make a simple pendulum mounted on a pivot so
that the simple pendulum can swing in a vertical plane. The weight is displaced
initially to an angle y0 from the vertical and released from rest. Now we want
to describe the motion of the simple pendulum in mathematical terms. Since
pendulum can move only in a vertical plane, the position of the pendulum can be
described completely by the knowledge of angle made by the rod from the vertical
position.
Let y(t) denote the angle made by the rod from the vertical position at a
given time t. To find the mathematical model we need to use an appropriate
physical rules governing the motion and simplifying approximations. In order
to determine the motion we need to impose some additional physical conditions
on the system. Here we assume the following assumptions:

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(i) The rod is rigid, of constant length l, and have no mass.
(ii) The weight is assumed to be a particle of mass m.
(iii) There is no other resistance to the motion of the pendulum. The only
external force present is the gravitational force acting on a point mass.

Since we are assuming that the pendulum was initially displaced to an angle
y0 and released from the rest then we also have y(0) = y0 and y 0 (0) = 0. At any
time t, the gravitational force F = mg is acting on the mass m in downward
direction. There is also a force of tension in the rod of magnitude T directed
along the rod towards the pivot. Then the equations governing the motion are
written as follows:
dy 2
−ml( ) = mg cos y − T, (1)
dt
d2 y
ml 2 = −mg sin y. (2)
dt
Since the motion of the pendulum is completely determined by the knowledge of
the function y(t), so only second equation is required i.e.

d2 y g
+ sin y = 0.
dt2 l
Definition 1. A differential equation is a relation between independent vari-
ables, dependent variables and its first or higher order derivatives.
In ordinary differential equation we have only one independent variable,
so that all derivatives of dependent variable present in differential are ordinary
derivatives and so differential equation is called an ordinary differential equation.
Example 2. Consider the following list of ordinary differential equations:
dy
1. dt = αy, α > 0;
d2 y
2. dt2 = g;
2
3. m ddt2y = mg − α dy
dt ;

d2 y k
4. dt2 + my = 0;
dy
5. dt + 5ty = t sin t;
2
6. t2 ddt2y + 3t dy
dt − 4y = 0;
2
7. (1 − t2 ) ddt2y − 2t dy
dt + n(n + 1)y = 0;
2
8. t2 ddt2y + t dy 2 2
dt + (t − n )y = 0.

2
 Each of the above equations is an ordinary differential equation. The
Definition 2. The order of a differential equation is the order of the highest
order derivatives present in the equation.
Equations (1) and (5) are first order and rest of the given equations are of
second order. Equation (1) represent a very basic population model where a
population is growing with the rate of α. Equation (2) represents an equation
of a particle of mass m falling freely under gravity while equation (3) may
represents the same model but now with the presence of air which exerts an
opposite force proportional to the velocity of the particle. Equation (4) may
represents a simple spring-mass problem. Equation (6) is a special type of
differential equation with variable coefficients known as Cauchy-Euler equation.
Equations (7) and (8) are classical examples known as Legendre’s equation and
Bessel’s equation respectively.

In partial differential equation there is more than one independent variable

and so the derivatives of dependent variables with respect to these independent
variable are partial derivatives rather than the ordinary derivatives.
Example 3. Let u(t, x, y) be a function of time t and the two rectangular co-
ordinates of a point in space x and y.
∂2u ∂2u
1. ∂x2 + ∂y 2 = 0;
2
∂2u
2. k( ∂∂xu2 + ∂y 2 ) = ∂u
∂t ;
2
∂2u ∂2u
3. k( ∂∂xu2 + ∂y 2 ) = ∂t2 ;

The above three equations are classical problems in partial differential equa-
tion and are widely used in applications. Equations (1)-(3) are classical in
nature and widely used in problems of fluid dynamics, theoretical physics and
many other related fields and are known as Laplace equation, 2-dimensional
heat equation, 2-dimensional wave equation respectively.

In this work we shall confine our attention to study only ordinary differential
equation and related tools.

Basic concepts
In our work, we always consider that the independent variable is real and is
denoted by t. The dependent variable for scaler equation is denoted by x and
for vector valued differential equations it is denoted by X. We also assume that
the differential equation and related functions are assumed to be real valued
functions. However, our results may be generalized, with minor modification,
for ordinary differential equations with complex case also.

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Definition 3. An ordinary differential equation of nth order is defined as follows
F (t, x, ..., xn ) = 0, (3)
i
here x , (i = 1, · · · , n) represent the ith derivatives of the unknown function x,
and F , defined in some subset of Rn+2 , denotes a relation between the (n + 2)
variables t, x, ..., xn .
Remark 1. Because of its implicit nature F (t, x, ..., xn ) = 0 equation (3) may
represents a collection of differential equations rather than a single differential
equation.
Example 4. Consider the following differential equation
(x0 )3 − 3t2 x02 + 3xx0 = 0.
It is given in the form (3) but it represent a combination of more than one
ordinary differential equations.
(x0 )3 − 3t2 (x0 )2 + 3xx0 = 0,
⇒ x0 ((x0 )2 − 3t2 x0 + 3x) = 0,
p
⇒ x0 = 0 or x0 = (3t2 ± 9t4 − 12x)/2.
So in order to avoid the ambiguity, we assume that given ordinary differential
equation is solvable in terms of the highest order derivative and written as in
the following form known as the normal form or the canonical form
xn = g(t, x, ..., xn−1 ). (4)
Definition 4. A function φ(t) is called a solution of (4) on t ∈ I := (a, b) if it
satisfies the following onditions
(i) φ(t) is defined and n times differentiable on I,
(ii) φ(t) satisfies the equation (4) for each t ∈ I.
The aim of the study of ordinary differential equation is to find the the
unknown function represented in an explicit form, preferably in terms of ele-
mentary function. In the absence of an explicit form, one need to study the
behavior of solutions by available analytical methods.

2 Classification
Before looking for a solution or any qualitative properties, we would like to iden-
tify the class in which the equation belongs to. There are various ways available
to classify the given ordinary differential equation. Some of the commonly used
classification are listed as follows.
• Classification based on function: Linear or Nonlinear
• Classification based on conditions: Initial value Problem (IVP ) or bound-
ary value problem (BVP).

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Linear and Non-linear Differential Equation:
Definition 5. Consider the differential equation (4). If the relation g is lin-
ear in its arguments x, · · · , xn−1 then the differential equation (4) is called a
linear ordinary differential equation otherwise it is called a nonlinear ordinary
differential equation.
Example 5.
x0 + kx = 0, k is a real constant (Linear).
dx
= sin x (Non-Linear).
dt
Let us consider the following differential equation of order two, written in
operator form:
L(x) := x00 + p(t)x0 + q(t)x = r(t),
here the notation L(x) suggest that the operator L operates on a function x to
give x00 + px0 + qx as its value.
Definition 6. An operator L : V (K) → V (K) is said to be a linear operator on
a vector space V defined on a scalar field K if it satisfies the following equality
L[αx + βy] = αL[x] + βL[y], ∀x, y ∈ V and ∀α, β ∈ K. (5)
Example 6. (i) x0 + kx = 0, k is a real constant.
(ii) x0 + a(t)x = b(t), a(t), b(t) are continuous functions defined on the interval
I.
(iii) a0 (t)xn + ... + an x = b(t), ai (t), i = 1, · · · , n are n known continuous
functions defined on I.
(iv)x0 + |x| = 0.,
(v) (x0 )2 + x = 0.
To check whether equation x0 + kx = 0 is linear or not, define
L[x] := x0 + kx
defined set of all continuously differential function over the field of real numbers.
Then we may easily check that the operator L satisfies the condition of linearity
(5) and hence the differential equation x0 + kx = 0 is a linear ordinary differ-
ential equation. Similarly we can see that the equations (i), (ii) and (iii) are
linear differential equation while equations (iv) and (v) are nonlinear differential
equation.
The reason behind this classification is that finding the explicit solution of
nonlinear differential equations are usually very difficult, if it is not impossible.
There are several methods available for solving linear differential equations but
no such general method are available for solving nonlinear differential equations.
Therefore, in the case of nonlinear differential equations, the methods which
provide the approximation solution or qualitative properties are very useful.

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Initial and Boundary Value Problem :
Consider the well known differential equation

x0 (t) + αx(t) = 0, t ∈ R (6)

which represents the population growth model in a single species. We may

easily check that x(t) = ceαt , where c is an arbitrary constant, is a solution of
the differential equation (6). Here, we get a one parameter family of solution
(consisting of infinitely many solution).
Frequently, we are interested only to find those solutions of (6) which also
satisfy certain other conditions. Such conditions may be represented in sev-
eral forms, but two of the important forms are initial conditions and boundary
conditions.
Example 7. For example, if we want to find out the population of the species
at any given time t provided that the population at time t0 was given as x0 .

(i) if we have x(t0 ) = x0 = 0 ⇒ c = 0 and the population x(t) will remain zero
for all future time t,
(ii) if x0 = 1, then c = e−αt0 and the population will be x(t) = eα(t−t0 ) .
(iii) if x(t0 = 0) = x(0) = x0 ⇒ c = x0 and then the population will be
x(t) = x0 eαt .

Example 8. Consider the following second order differential equation

x00 + (p + q)x0 + pqx = 0. (7)

The solution of the given equation is x(t) = αe−pt + βe−qt . If x(0) = 0, x0 (0) =
q − p. Then using the given conditions we have α = 1, β = −1, so the particular
solution satisfying the given condition is given as x(t) = e−pt − e−qt .
Example 9. Consider the following differential equation for the motion of sim-
ple pendulum:
x00 + x = 0, (8)
solution of the given equation is x(t) = α sin t + β cos t. Where t ∈ R and α, β
are arbitrary constant.
(a) If x(0) = 0, x(π/2) = 0, ⇒ x(t) = 0.
(b) If x(0) = 0, x(π/2) = 1, ⇒ x(t) = sin t.
Note that in example (8), conditions are given at one point while in exam-
ple (9) conditions are given at two different points. Conditions given at the
same value of t are known as initial conditions while the conditions defined at
two (generally at the end point of interval) or more different points are called
boundary conditions.

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Definition 7. The equation along with initial conditions is known as initial
value problem (IVP). Similarly, a differential equation along with boundary con-
ditions is known as boundary value problem (BVP).
Remark 2. Consider the following differential equations

(i) 2(x0 )2 + t2 = 0 does not have a real valued solution.

(ii) x00 + |x| = 0 , 0 ≤ t ≤ π, with x(0) = 0 , x(0) = A. We may find the
conditions on A such that the boundary value problem has no solution,
unique solution, and infinitely many solutions. Thus a boundary value
problem may or may not have any solution or may have more than one
solutions under different conditions.
(iii) An initial value problem may have no solution, only one solution or may
have more than one solution. For example
(a) The initial value problem tx0 − 3x + 3 = 0, x(0) = 0 has no real
solution,
(b) The initial value problem tx0 − 3x + 3 = 0, x(1) = 1 has one and only
one real solution x(t) ≡ 0 of the differential equation, and
1
(c) Initial value problems tx0 −3x+3 = 0, x(0) = 1 and x0 = x 2 , x(0) = 0,
have more than one solution(infinitely many!).

3 The need of theory

It is observed that differential equations are usually originated as an effort of
creating a mathematical model for the motion of physical system such as sim-
ple pendulum, or simple spring-mass problem. While modeling of the physical
problem we may use different physical approximations and in result of this we
may lead to different differential equations. Recall that the differential equa-
tion for the simple pendulum starting from an initial angle y0 is a nonlinear
differential equation given as follows

d2 y g
+ sin y = 0. (9)
dt2 l
Now, we want to use equation (9) to find the motion of the pendulum satisfying
the initial conditions y(0) = y0 , y 0 (0) = 0, and we are surprised to know that
there is no method available to solve the differential equation (9) in terms of
elementary functions. But in real world a simple pendulum actually moves, this
means our mathematical model is useless and have certain problems with the
set of assumptions and we need to construct a new model for the same so that
the new model have some solutions.

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 Also, we have seen differential equations which may not have any solution,
but such equations may not be so important. Many physical systems are hav-
ing real solutions, whether a suitable mathematical models is available or not.
Therefore, to construct a useful mathematical model, we need to know that
whether it has a solution or not. Thus the existence problem i.e. identi-
fying the class of differential equation which admits solution is indeed a very
important problem of mathematical theory.
Having conditions about the existence of solution is not the only requirement
that is desirable for a useful model. Since the problem relating to differential
equation are originated from physical systems and expected to have unique
solution for the given set of conditions. There are many differential equations
which do not have unique solution(see Remark(2)). Thus we refer this problem
i.e. finding the conditions such that a differential equation should have exactly
one solution for a given set of initial conditions as a uniqueness problem.
One of the important property regarding the satisfactory model is that if
the initial conditions are slightly changed then we expect that the outcomes are
also slightly changed and it it is desirable the our mathematical model should
also have this important property i.e. the solutions of the differential equations
depends continuously on the initial conditions. We refer to this property as
continuity of the solution on initial conditions.
Example 10. Consider the differential equation

x0 = x, with the initial condition (10)

x(0) = 0. (11)

The solution of the given initial value problem is the trivial solution x(t) ≡ 0.
Now if we slightly perturb the initial condition (11) and take x(0) = ,  > 0.
The solution of the perturbed problem is now x(t) = et . The trivial solution is a
bounded solution but the solution of the same differential equation with slightly
perturbed initial condition will be unbounded as t → +∞.
Now we observe that a slight change in initial condition change the nature
of the solution. Thus, a mathematical model of a physical system should have
the following properties:
1. A solution should exist satisfying the initial conditions.

2. There exists a unique solution corresponding to each set of initial condi-

tions.
3. Solutions of the differential equations depends continuously on the initial
conditions.
The above said conditions are known as well-posed ness conditions for a math-
ematical model and a mathematical problem having these properties is called a
wellposed problem.

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 Finding wellposed mathematical models is a rare phenomena and even for
wellposed problems, finding the solution in its explicit form is really very diffi-
cult. There are several equations whose solution are not explicitly known have
importance in many areas of science and technology.
Example 11. x00 + p(t) = 0 is an innocent looking equation which can not be
solved for a general choice of p(t), yet is has many useful applications.
So for these problems we study the nature of solutions through analytical
considerations. The properties of solutions like existence, uniqueness, continua-
tion of solution, dependence of initial data, bounded-ness, stability, periodicity,
asymptotically behavior, etc. provide the nature and behavior of solutions of
these problems.