Differential Equation

MAT223

CHAPTER 1:
INTRODUCTION TO D.E

• Basic definition and terminology
• The existence and uniqueness theorem
• Phase lines and direction fields
• Differential equations as mathematical models

1

Definition and terminology

Definition 1.1: Differential Equation

An equation containing the derivatives of one or more dependent variables, with respect to one
or more independent variables, is said to be a differential equation (DE).

Example: Recall chain rule
We have a function = 0.1 2,

By the chain rule, its derivative is

= 0.2 0.1 2 Classifying
differential
equations

Replacing = 0.1 2 into the derivative, • Type
• Order
we will get • Linearity

Differential
= 0.2 equation

2

Classification by type:

Ordinary differential equation Partial differential equation
(ODE) (PDE)

• Contains only ordinary derivatives of one or more • Involves partial derivatives of one or more
dependent variables with respect to a single dependent variables of two or more independent
independent variable. variable.

Example: Example:

1. + 5 = , 1. 2 + 2 = 0,
2 2


2. 2 − + 6 = 0, 2. 2 = 2 − 2
2 2 2

3. + = 2 + 3. = −



3

Classification by order:

• The order of a differential equation (either ODE or PDE) is the order of the highest derivative in
the equation

Example:

2 + 5 3 − 4 = 0,
2

= , , ′, ′′, … −1



= ,

2 = , , ′
2

4

Classification by linearity: Nonlinear differential equation Is this equation linear or
nonlinear? Why?
Linear differential equation • A nonlinear ODE is simply one that is not
linear. ′ + 2 = sin
• A combination of some unknown
function and its derivatives with • Nonlinear functions of the dependent Write one linear DE:
some coefficients which are not variable or its derivatives, such as sin
constants, but now functions of the or , cannot appear in linear equation. Write one nonlinear DE
independent variable .

Example: Example:
1. ′ + = 1. ′′ 2 =
2. ′ = −
2. ′′ + ′ + = 3. ′ = sin

5

Solution of an Ordinary Differential Equation

Definition 1.2: Solution of an ODE

Any function Φ, defined on an interval , and possessing at least derivatives that are
continuous on , which when substituted into an nth-order ordinary differential equation
reduces the equation to an identity, is said to be a solution of the equation on the interval.

, Φ , Φ′ , … , Φ = 0, for all in

Interval • Interval of definition • Open interval ( , )
• Interval of validity • Closed interval [ , ]
• Domain of solution • Infinite interval ( , ∞)
• Interval of existence • Etc…

6

Example 1.1: Verification of a solution

Verify that the indicated function is a solution of the given differential equation on the interval

−∞, ∞ ?

= 1/2; = 1 4.
16

Solution: What can you say about
Check if each side of the equation is the same for every in the interval.
this equation?
= 1/2 ′′ − 2 ′ + = 0;
= .

LHS RHS

LHS: = 1 4 ∙ 3 = 1 3,
RHS: 16 4
1

1/2 = ∙ 1 4 2 = ∙ 1 2 = 1 3.

16 4 4

Each side of the equation is the same for every real number .

7

Solution of an Ordinary Differential Equation

Definition 1.3: Implicit Solution of an ODE

A relation , = 0 is said to be an implicit solution of an ordinary differential equation

, , ′, … , of an interval , provided that there exists at least one function Φ that

satisfies the relation as well as the differential equation on .

, = 0
, Φ( ) = 0

Example 1.2: Verification of an implicit solution
Verify that the relation 2 + 2 = 25 is an implicit solution to the differential equation


= − ,
on an open interval (−5,5).

8

Solution:

Solving 2 + 2 = 25 for in terms of yields = ± 25 − 2.

= Φ1 = 25 − 2 satisfy the relation that 2 + Φ12 = 25 and 2 + Φ22 = 25 and are the explicit solutions
Φ2 =− 25 − 2

defined on the interval (−5,5).

From here, we will generate:



55

−5 5 −5 5 −5 5

−5 Explicit solution: −5
1 = 25 − 2
Implicit solution: Explicit solution:
= ± 25 − 2 −5 < < 5 1 = − 25 − 2

−5 < < 5

9

Families of solutions

One-parameter family -parameter family Free of parameters

A solution of ( , , ′) = 0 A solution of A solution of differential
containing a constant is a ( , , ′, … , ( )) = 0 equation that is free of
set of solutions ( , , ) = parameters is called a
containing a constant is a
0. particular solution.
set of solutions
( , , 1, 2, … , ) = 0.

10

Initial-Value Problems

• Sometimes in physics we are not interested in all solutions to a differential
equation, but only in those solutions satisfying extra conditions.

• For example, in the case of Newton’s second law of motion for a point particle, we
could be interested only in solutions such that the particle is at a specific position at
the initial time.

• Such condition is called an initial condition, and it selects a subset of solutions of the
differential equation.

• An initial value problem means to find a solution to both a differential equation and
an initial condition.

11

On some interval containing 0, the problem of solving an -th order differential
equation subject to side conditions specified at 0:

Solve : = , , ′, … , ( −1)
Subject to
: 0 = 0, ′ 0 = 1, … , −1 0 = −1
-th order initial
value problem

Arbitrary constants

The values of ( ) and its first − 1 derivatives at
0, 0 = 0, ′ 0 = 1, … , −1 0 = −1

are called the initial conditions.

12

Existence and Uniqueness Existence

Theorem 1.1: Existence and • Does the differential equation = , possess
uniqueness for a linear DE
solutions?
If the functions 0 , 1 , … , ( ) and ( )
are continuous on the interval and 0 ≠ • Do any of the solution curves pass through the point
0 ∀ ∈ , then the initial value problem 0, 0 .

+ −1 −1 + ⋯ + 1 ′ Uniqueness
+ 0 = ( )
• When can we be certain that there is precisely one
where 0 = 0, ′ 0 = 1, … , ′′ 0 = solution curve passing through the point 0, 0
−1, 0 ∈

has a unique solution on the interval .

13

Example 1.3:
Does ′′ − 4 = 0 satisfying the initial conditions 0 = 1, ′ 0 = 2 has a unique

solution on the interval −∞, ∞ ?

Solution:
In ′′ − 4 = 0 , we obtain that

0 , = −4, 1 = 0, 2 = 1 and = 0
are all continuous on the interval −∞, ∞ and 0 ≠ 0, ∀ ∈ −∞, ∞ .

Therefore, the IVP has a unique solution on the interval −∞, ∞ .

It can be shown that = 2 is the unique solution. It is all continuous on the interval

−∞, ∞ but 0 = 0 at = 0 and 1 ∈ 0, ∞ . Hence the largest interval is 0, ∞ .
2

Remark: 14
1. A unique solution may still exist even though the conditions of the theorem are not satisfied.
2. A boundary value problem does not necessarily have a solution. In addition, when a solution of this type of

problem exists, it is not necessarily unique.

Theorem 1.2: Existence of a unique
solution

Consider the IVP = , , 0 = 0.



Let be a rectangular region in plane
defined by ≤ ≤ , ≤ ≤ that
contains the point 0, 0 in its interior.

( 0, 0)

If , and are continuous on , then

0 − ℎ 0 + ℎ
there exists some interval

0: 0 − ℎ < < 0 + ℎ, ℎ > 0

contained in ≤ ≤ , and a unique function

( ), defined on 0, that is a solution of the
given IVP.

15

Phase lines

• A phase line is a diagram that shows the qualitative behaviour of an autonomous ODE in

a single variable = .


• It indicates the values of the dependent variable for which is increasing, decreasing or
constant

Algorithm For Drawing A Phase Line

i. Draw a horizontal line
ii. Find the equilibrium points (i.e. values such that ( ) = 0) and mark them on the line
iii. Find intervals for which ( ) > 0 and mark them with up arrows → or >
iv. Find intervals for which ( ) < 0 and mark them with down arrows  or <
v. Repeat steps iii and iv for second derivative ′ .

16

Example 1.4:

Draw a phase line for the equation

= + 1 ( − 2)

and use it to sketch solutions to the equation.

Steps i and ii:
Draw a number line for and mark the equilibrium values = − and = where / = .



-1 2

Steps iii and iv:
Identify and label the intervals where ’ > and ’ <

’ > 0 ’ < 0 ’ > 0 ’
−2 4>0

-2 -1 0 23 0 −2 < 0

3 4>0

-1 2 17

Step v:
Calculate ” and mark the intervals where ” > and ” <
We differentiate ’ with respect to using implicit differentiation

y ' = ( y + 1)( y − 2) = y 2 − y − 2

( ) ( )y" = d y ' = d y 2 − y − 2
dx dx
= 2 yy' − y'
= (2 y −1)y '
= (2 y −1)( y + 1)( y − 2)

We see that ” changes sign at = −1, = 1/2, and = 2.

” < 0 ” > 0 ” < 0 ” > 0 ′′
−2 −20 < 0
-2 -1 0 1/2 1 2 3 0
1 2>0
3 −2 < 0
20 > 0

18

Final step:
Sketch an assortment of solution curves in the -plane

19

Direction fields

• With a simple concept from calculus, a derivative of a differentiable function =


gives slopes of tangent lines at points on its graph.


= ,
• , in the equation is called the slope function.

• A slope field (direction field) is a graphical representation of the solutions of a first order
differential equation.

• It is achieved without solving the differential equation analytically.

• The representation may be used to qualitatively visualize solutions, or to numerically
approximate them.

20

Example 1.5:
Given a differential equation

−
=

Create a table that satisfies the differential equation:




01 0

11 -1

1 0 undefined

-1 -1 -1

1 -1 1

22 -1

-2 2 1

2 -2 1

-2 -2 -1 21

Other examples:

22

Differential equations as mathematical models

What is mathematical model?
➢ An idealization of the real-world phenomenon without a complete accurate representation
➢ Help us understand a behaviour better and aid us in planning for the future
➢ Allows us to reach mathematical conclusions about the behaviour
➢ These conclusions can be interpreted to help decision maker plan for the future

Definition 1.4: Mathematical Model

A representation in mathematical terms of the behaviour of real devices and objects

23

Steps of the modelling process with differential equations

If necessary, alter Assumptions Express
assumptions or & assumptions in
increase resolution
Hypothesis terms of DE
of model
Mathematical
Check model formulation
predictions with

known facts

Display model Solve the DEs
predictions
Obtain solutions

24

Objectives of mathematical model

1) Developing • How to generate mathematical
scientific representations or models?

understanding

2) Test the effect of • How to use and validate the models?
changes in system

3) Aid decision • Tactical decisions by managers
making, including • Strategic decisions by planners
• How and when their use is limited

25

Mathematical model for physical systems

Population
dynamics

Spread of Chemical
disease reactions

Newton’s Law Mathematical Radioactive
of cooling model decay
or warming

Mixtures Series of
circuits

Falling
bodies

26