ORDINARY DIFFERENTIAL EQUATION

Noorsaadah binti Ishak



Writer: Noorsaadah binti Ishak
Editor: Norhafizah binti Arshad

First Printing 2022

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National Library of Malaysia Cataloguing-in-Publication Data

Noorsaadah Ishak, 1985-
STEP BY STEP APPROACH : ORDINARY DIFFERENTIAL EQUATION /
Writer : Noorsaadah binti Ishak ; Editor : Norhafizah binti Arshad.
Mode of access: Internet
eISBN 978-967-2860-32-7
1. Differential equations.
2. Differential equations--Numerical solutions.
3. Government publications--Malaysia.
4. Electronic books.
I. Norhafizah Arshad, 1984-.
II. Title.
515.352

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PREFACE

ABSTRACT

iv 6

“

”

PREFACE iii
iv
ABSTRACT v
1
TOPIC OUTLINE 2
7
DEFINITION 14
45
01 ORDER AND DEGREE 54
02 FORMATION OF ORDINARY

DIFFERENTIAL EQUATION

03 FIRST ORDER DIFFERENTIAL
EQUATION

04 SECOND ORDER DIFFERENTIAL
EQUATION
REFERENCE

Ordinary Differential Equation is an
equation which consists of one or
more functions of one independent

variable along with their derivatives.

equation
+ =

function
Differential/derivative

ORDER & DEGREE

The order of the Order 1
highest derivative

(also known as = 0
differential coefficient)
present in the equation 2 3 4
2 + 3 = 6

Order 2

Degree 3

2 3 4
2 + 3 = 6 Represented by the
highest power of the
4 2 2 2 4 highest order derivative
2 2 present in the equation
= − 2

Degree 4

TRY THIS!

State the order and the degree of
1 the following equation
4 2 = −
2
−

2 7+ 3 = 4 2
3 3 4

5 3 + 2 = − 5


4 5 3 = 2 log
3
−

FORMATION OF ODE

Consider the equation
= + where is a constant

and is the parameter. The
differential equation of one
parameter family is obtained by
differentiating the equation of the
family once and by eliminating the
parameter from the equation.

FORMATION OF ODE

Determine the 01
number of parameter

02 Differentiate the
equation equals to the
number of parameter

Use suitable method to 03
form the new equation

by eliminating all
parameters

Q1 Form the differential
equation of the curve

= 20 2 + 5

Step 1
Determine the number of parameter
= 20 2 + 5

1 parameter

Step 2
Differentiate the equation equals to the number of
parameter

= 20 2 + 5
= 40



Step 3
Use suitable method to form the new equation by
eliminating all parameters

= 40 No more parameter



Q2 Form the differential
equation of the curve
= cos 5 − sin 5

Step 1
Determine the number of parameter

= cos 5 − sin 5

2 parameters

Step 2
Differentiate the equation equals to the number
of parameter

= cos 5 − sin 5

= −5 sin 5 − 5 cos 5



2 = −25 cos 5 + 25 sin 5
2

Step 3
Use suitable method to form the new equation by
eliminating all parameters

2 = −25 cos 5 − sin 5
2

2 = −25 No more parameter
2

Q3 Form the differential
equation of the curve

= 3 3

Step 1
Determine the number of parameter
= 3 3

1 parameter

Step 2
Differentiate the equation equals to the number
of parameter

= 3 3

= 9 3


Step 3
Use suitable method to form the new equation by
eliminating all parameters

= 3[3 3 ]


2 = 3 No more parameter
2

Q4 Form the differential
equation of the curve

= 5 − 2

Step 1
Determine the number of parameter
= 5 − 2

2 parameters

Step 2
Differentiate the equation equals to the number
of parameter

= 5 − 2

= 5 − 2


2 = −2
2

Q4 Form the differential
equation of the curve

= 5 − 2

Step 3 Use suitable method to form the new
equation by eliminating all parameters

2 = −2 , then − 1 2 =
2 2 2

= 5 − 2 , then


= 5 − 2[− 1 2 2 ]
2

= 5 + 2
2

− 2 = 5
2

= 5 − 2, then

= − 2 − [− 1 2 2 ] 2
2 2

= − 2 2 + 2 2
2 2 2

= − 2 2 No more parameter
2 2

TRY THIS!

Form the differential equation of
the following curve:

1
= +

2 ANSWER:

= −4 +

3 ANSWER:

= sin 2

4 ANSWER:

= 2 (2 − 3) 4

ANSWER:

FIRST ORDER OF ODE

1 2

DIRECT
INTEGRATION

3 SEPARABLE
VARIABLE

SUBSTITUTION/ 4
HOMOGENEOUS

INTEGRATING
FACTOR

An ordinary differential equation

of = ( ) can be solved by

integrating both sides with
respect to x; ‫ ׬‬ = ‫ ׬‬ ( ) .

This technique, called DIRECT
INTEGRATION, can also be

applied when the left hand side is
a higher order derivative.

DIRECT INTEGRATION

Split the derivative, 01
one side for dy,

another side for dx

02 Rearrange the
variables according to

the derivative

Integrating both sides 03

Q1 Solve the first order
differential equation of

= 5 3 − 5


Step 1 Split the derivative
= (5 3− 5)

Step 2 Rearrange the variables according to the
derivative

= (5 3− 5)

Consist only 1 variable.
No need to rearrange

the position.

Step 3 Integrating both sides

න = න(5 3− 5)
5 4 6

= 4 − 6 +

1Solve the first order
differential equation of
Q2 3
= 5 5


Step 1 Split the derivative

1 = 5 5
3

Step 2 Rearrange the variables according to the
derivative
Properties of indices,

× = +

= 5 5 3

= 5 8

Step 3 Integrating both sides
‫ ׬‬ = ‫ ׬‬5 8
= 5 8 +

8

Solve the first order
differential equation of

Q3
sin 2 = 1

Step 1 Split the derivative
sin 2 = 1

Step 2 Rearrange the variables according to the
derivative

sin 2 = 1 Consist only 1 variable.
No need to rearrange
the position.

Step 3 Integrating both sides

‫ ׬‬sin 2 = ‫ ׬‬1

− cos 2 = +
2

TRY THIS!

Solve the first order differential equation below:

1 2
2 = 3

3 ANSWER:

= ( 2−3) 3


ANSWER:

5
=

4 ANSWER: 1 = 3
ANSWER:

A separable differential equation is
any differential equation that we can

write in the following form:


= ( )
You will have to separate the
function into 2 parts with different
variables and integrate both parts
accordingly. The new separable form
can be write as =

SEPARABLE METHOD

Split the derivative, 01
one side for dy,

another side for dx

02 Rearrange the
variables according to

the derivative

Integrating both sides 03

Q1 Solve the first order
differential equation of

= 5 3 − 5


Step 1 Split the derivative
4 3 = 6

Step 2 Rearrange the variables according to the
derivative

4 3 = 6



4 2 = 6

Step 3 Integrating both sides
‫ ׬‬4 2 = ‫ ׬‬ 6
4 3 = 7 +

37

Q2 Solve the first order
differential equation of

2 = 5 5
4

Step 1 Split the derivative

2 = 5 5
4

Step 2 Rearrange the variables according to the
derivative

1 = 5 5
4 2

−4 = 5 4
2

Step 3 Integrating both sides

‫׬‬ −4 = ‫׬‬ 5 4
2

−4 = 5 5 +
−4 2×5

− −4 = 5 +

42

Solve the first order
differential equation of

Q3
( − 1) = 7

Step 1 Split the derivative
− 1 = 7

Step 2 Rearrange the variables according to the
derivative

= 7

( −1)

Step 3 Integrating both sides

‫ ׬‬ = ‫׬‬ 7
( −1)

2 = 7 ln( − 1) +

2

TRY THIS!

Solve the differential equation below:

1 2 = 3
2
3 ANSWER:
4

= 2 − 1

ANSWER:


= sin

ANSWER:

2 4
= 2 −

ANSWER:

An equation is said to be
homogeneous if it can be written

in the form
= ( )



You need to substitute:

= +
=

HOMOGENEOUS METHOD

Substitute the formula 01

= +
=

02 Bring v to the right side.
Simplify the right sided

function

Split dv to the left side and 03
dx to the right side.

Rearrange the variables
accordingly

04 Integrating both sides

Substitute = 05



Q1 Solve the first order
differential equation of


( + ) =

Step 1 Substitute the formula

Step 1 Substitute the formula

= =( + ( +) )
+ + = (= + ( + ) )

Step 2 Bring v to the right side. Simplify the
right sided function

Step 2 Bring v to the right side. Simplify
t h+e r i g h=t s i d ed function
1+

+ = =

− 1+
1+

= = − 1 1(++1 + )−

= = −1 + −− 1 2(+1 + )

= =− − 2 − 2
11++

= − 2

1+

Q1 Solve the first order
differential equation of


( + ) =

Step 3 Split dv to the left side and dx to the right side.
Rearrange the variables accordingly
= − 2

1+

− (1+ ) = 1
2

Step 4 Integrating both sides

‫׬‬ − (1+ ) = ‫׬‬ 1
2

‫׬‬ − 1 − = ln +
2 2

‫׬‬ − −2 − 1 = ln +


− −1 − ln = ln +

−1

1 − ln = ln +



Step 5 Substitute =



1 − ln = ln +





− ln = ln +


Solve the first order

Q2 2 −
differential equation of

2 + 2 =0

Step 1 Substitute the formula

Step 1 Substitute the formula
2 = 2 + 2
=
(= +( )2+2 2)
=2

+ (= + 2 + )2( )2


+
2 ( )

Sritgeh p t+2si d B er di n=fgu v n2+2ct 2t o i o22t n h2e right side. Simplify the

+ = 2 1+2 2
+ = 2 2

1+
1+2 2
+ =
= 2

1+ −

= − (1+ )

S tep 21+ B ring v to the right side. Simplify
t h =e r −ig −h t 2 sided function

1+

= 1+2 2 −
= − 2 2

= 11++2 2−2 2

2

= 1

2

Solve the first order

Q2 2 −
differential equation of

2 + 2 =0

Step 3 Split dv to the left side and dx to the right side.
Rearrange the variables accordingly

= 1
2

2 = 1



Step 4 Integrating both sides

‫׬‬ 2 = ‫׬‬ 1


2 2 = ln +

2

2 = ln +

Step 5 Substitute =



( )2= ln +



2 = ln +
2

Solve the first order
differential equation of

Q3 − = ( + 3 )

Step 1 Substitute the formula

Step 1−Su b s t it=ut e+th3e f o rmula

= +−3( + = )


+ = = −

+3( + )

+ = −
Step 2 B r i ng v +3t(o t)he right side. Simplify the
righ t+si d e d =fu n ((1c1+t−3i o ))n

+ + = = 1 −

11++ 3

= −

S tep1+2 Bring v to the right side. Simplify
t h =e r −ig (h1+t s)ided function

1+

== −111+ −+ −3 2−
==−1− −12+ (31 +3 )

1+

= 1− − −3 2

1+3

= 1−2 −3 2

1+3

Solve the first order
differential equation of

Q3 − = ( + 3 )

Step 3 Split dv to the left side and dx to the right side.
Rearrange the variables accordingly

= 1−2 −3 2

1+3

1+3 = 1
1−2 −3 2

Step 4 Integrating both sides Use substitution method to
solve the integral
‫׬‬ 1+3 = ‫׬‬ 1
1−2 −3 2 Let = 1 − 2 − 3 2
= −2 − 6



=

−2(1+3 )

− 1 ln(1 − 2 − 3 2) = ln + = ‫׬‬ 1+3
2 1−2 −3 2

=‫׬‬ 1+3
−2(1+3 )

=‫׬‬ 1
−2

= 1 ln +
−2

= 1 ln(1 − 2 − 3 2) +
−2

Step 5 Substitute =



− 1 ln(1 − 2 − 3( )2) = ln +

2

− 1 ln(1 − 2 − 3 22) = ln +
2

TRY THIS!

Solve the differential equation below:

1 = 2 2+
2

3
ANSWER:

2 + + = 0

ANSWER:

2 + 2 − = 0

4 ANSWER:

2 3 + 3 − 3 2 = 0

ANSWER:

Suppose we have the first order

differential equation + =



where P and Q are functions
involving only. We can solve these

differential equations using the
technique of an integrating factor

INTEGRATING FACTOR

Express the equation in 01
the form of

+ =

Find P and Q

02 Find IF where = ‫ ׬‬

Substitute into formula 03
∙ = න ∙

Identify the functions P(x)
and Q(x) in the following

differential equations.

+ 3 = 2



+ 2 cos =

′ + 5 = 3
2

5
= + 3

5
= + 3

Q1 Solve the first order
differential equation of

1
+ = 3

Step 1 Express the equation in the form of

+ = . Find P and Q.


+ 1 = 3



= 1 , = 3



Step 2 Find IF where = ‫ ׬‬
IF = ‫׬‬ 1
IF = ln
IF =

Step 3 Substitute into formula ∙ = ‫ ׬‬ ∙

∙ = ‫ ׬‬ ∙
∙ = ‫ ׬‬3 ∙
= 3 2 +

2

Q2 3 Solve the first order
differential equation of

− 2 = 5


Step 1 Express the equation in the form of
+ = . Find P and Q.



3 − 2 = 5


1 5
− = 3

15
= − , = 3

Step 2 Find IF where = ‫ ׬‬

IF = ‫ ׬‬− 1
IF = − ln = ln −1
IF = −1 = 1



Q2 3 Solve the first order
differential equation of

− 2 = 5


Step 3 Substitute into formula ∙ = ‫ ׬‬ ∙

∙ = ‫ ׬‬ ∙

∙ 1 = ‫׬‬ 5 ∙ 1
3

= ‫׬‬ 5
4

= ‫׬‬ 5 −4


= 5 −3 +
−3

= − 5 +
3 3

Solve the first order
differential equation of
Q3
− 4 = 5 2

Step 1 Express the equation in the form of

+ = . Find P and Q.



+ 4 = 2


= 4, = 2

Step 2 Find IF where = ‫ ׬‬
IF = ‫ ׬‬4
IF = 4

Step 3 Substitute into formula ∙ = ‫ ׬‬ ∙
∙ = ‫ ׬‬ ∙
∙ 4 = ‫ ׬‬ 2 ∙ 4
∙ 4 = ‫ ׬‬ 6
∙ 4 = 6 +

6