CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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Classification by type
- Ordinary Differential Equations (ODE)
 Contains one or more dependent variables
 with respect to one independent variable
is the dependent variable
while is the independent
variable
is the dependent variable
while is the independent
variable
Dependent Variable: u
Independent Variable: t
- Partial Differential Equations (PDE)
 involve one or more dependent variables
 and two or more independent variables
Can you determine which one is the DEPENDENT VARIABLE and which
one is the INDEPENDENT VARIABLES from the following equations ???
Dependent Variable: w
Independent Variable: x, t
Dependent Variable: u
Independent Variable: x, y
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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Classification by order / degree
- Order of Differential Equation
 Determined by the highest derivative
- Degree of Differential Equation
 Exponent of the highest derivative
Examples:
a)
Order : 1
Degree: 2
b)
Order : 2
Degree: 1
c)
Order : 2
Degree: 1
d)
Order : 3
Degree: 4
Classification as linear / nonlinear
- Linear Differential Equations
 Dependent variables and their derivative are of degree 1
 Each coefficient depends only on the independent variable
 A DE is linear if it has the form
Examples:
1)
2)
3)
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- Nonlinear Differential Equations
 Dependent variables and their derivatives are not of degree 1
Examples:
1)
2)
3)
Order :
1
Degree: 1
Order :
1
Degree: 2
Order :
3
Degree: 2
Initial & Boundary Value Problems
Initial conditions : will be given on specified given point
Boundary conditions : will be given on some points
Examples :
1)
Initial condition
2)
Boundary condition
Initial Value Problems (IVP)
Initial Conditions:
Boundary Value Problems (BVP)
Boundary Conditions:
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Solution of a Differential Equation
- General Solutions
 Solution with arbitrary constant depending on the order
of the equation
- Particular Solutions
 Solution that satisfies given boundary or initial conditions
Examples:
(1)
Show that the above equation is a solution of the following DE
(2)
Solutions:
(3)
(4)
Insert (1) and (4) into (2)
Proven that
is the solution for the given DE.
EXERCISE:
Show that
is the solution of the
following DE
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Forming a Differential Equation
Example 1:
Find the differential equation for
Solution:
(1)
(2)
Try to eliminate A by,
a) Divide (1) with
:
(3)
b) (2)+(3) :
(4)
Example 2:
Form a suitable DE using
Solutions:
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Exercise:
Form a suitable Differential Equation using
Hints:
1. Since there are two constants in the general solution,
be differentiated twice.
2. Try to eliminate constant A and B.
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
1.2 First Order Ordinary Differential Equations (ODE)
Types of first order ODE
- Separable equation
- Homogenous equation
- Exact equation
- Linear equation
- Bernoulli Equation
1.2.1 Separable Equation
How to identify?
Suppose
Hence this become a SEPARABLE EQUATION if it can be written as
Method of Solution : integrate both sides of equation
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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Example 1:
Solve the initial value problem
Solution:
i)
Separate the functions
ii)
Integrate both sides
Use your Calculus
knowledge to
solve this
problem !
Answer:
iii)
Use the initial condition given,
iv)
Final answer
Note: Some DE may not appear separable initially
but through appropriate substitutions, the DE can be
separable.
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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Example 2:
Show that the DE
can be reduced to a separable
equation by using substitution
. Then obtain the solution
for the original DE.
Solutions:
i)
Differentiate both sides of the substitution wrt
 (1)
 (2)
ii)
Insert (2) and (1) into the DE
 (3)
iii)
Write (3) into separable form
iv)
Integrate the separable equation
Final answer :
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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Exercise
1) Solve the following equations
a.
b.
c.
d.
2) Using substitution
, convert
to a separable equation. Hence solve the original equation.
1.2.2 Homogenous Equation
How to identify?
Suppose
,
is homogenous if
for every real value of
Method of Solution :
i)
Determine whether the equation homogenous or not
ii) Use substitution
and
in the original DE
iii) Separate the variable and
iv) Integrate both sides
v) Use initial condition (if given) to find the constant value
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Separable
equation
method
CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
Example 1:
Determine whether the DE is homogenous or not
a)
b)
Solutions:
a)
this differential equation is homogenous
b)
this differential equation is non-homogenous
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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Example 2:
Solve the homogenous equation
Solutions:
i) Rearrange the DE
 (1)
ii) Test for homogeneity
this differential equation is homogenous
iii) Substitute
into (1)
and
iv) Solve the problem using the separable equation method
Final answer :
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Note:
Non-homogenous can be reduced to a homogenous
equation by using the right substitution.
Example 3:
Find the solution for this non-homogenous equation
(1)
by using the following substitutions
(2),(3)
Solutions:
i)
Differentiate (2) and (3)
and substitute them into (1),
ii)
iii)
Test for homogeneity,
Use the substitutions
and
REMEMBER!
Now we use
instead of
iv)
Use the separable equation method to solve the
problem
LASTLY, do not forget to
Ans:
replace
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1.2.3 Exact Equation
How to identify?
Suppose
,
Therefore the first order DE is given by
Condition for an exact equation.
Method of Solution (Method 1):
i)
Write the DE in the form
And test for the exactness
ii)
If the DE is exact, then
(1), (2)
To find
, integrate (1) wrt
to get
(3)
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
iii)
To determine
iv)
v)
Integrate
to get
Replace
into (3). If there is any initial conditions given,
substitute the condition into the solution.
Write down the solution in the form
, where A is a constant
vi)
, differentiate (3) wrt
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to get
Method of Solution (Method 2):
i)
Write the DE in the form
And test for the exactness
ii)
If the DE is exact, then
(1), (2)
iii)
To find
from
, integrate (1) wrt
to get
(3)
iv)
To find
from , integrate (2) wrt
to get
(4)
v)
vii)
viii)
ix)
Compare
and
to get value for
and
.
Replace
into (3) OR
into (4).
If there are any initial conditions given, substitute the conditions into
the solution.
Write down the solution in the form
, where A is a constant
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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Example 1:
Solve
Solution (Method 1):
i) Check the exactness
Since
, this equation is exact.
ii) Find
(1)
(2)
To find
, integrate either (1) or (2), let’s say we take (1)
(3)
iii) Now we differentiate (3) wrt to compare with
(4)
Now, let’s compare (4) with (2)
iv) Find
v) Now that we found
, our new
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should looks like this
CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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vi) Write the solution in the form
, where
is a constant
Exercise :
Try to solve Example 1 by using Method 2
Answer:
i)
Check the exactness
Since
ii)
, this equation is exact.
Find
(1)
(2)
To find
, integrate both (1) and (2),
(3)
(4)
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
iii)
Compare
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to determine the value of
and
Hence,
and
iv)
Replace
into
OR
v)
Write the solution in the form
into (4)
Note:
Some non-exact equation can be turned into exact
equation by multiplying it with an integrating factor.
Example 2:
Show that the following equation is not exact. By using integrating
factor,
, solve the equation.
Solution:
i) Show that it is not exact
Since
, this equation is not exact.
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
ii) Multiply
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into the DE to make the equation exact
iii) Check the exactness again
Since
, this equation is exact.
iv) Find
(1)
(2)
To find
, integrate either (1) or (2), let’s say we take (2)
(3)
v) Find
(4)
Now, let’s compare (4) with (1)
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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vi) Write
, where
Exercises :
1. Try solving Example 2 by using method 2.
2. Determine whether the following equation is exact. If it is,
then solve it.
a.
b.
c.
3. Given the differential equation
i. Show that the differential equation is exact. Hence, solve
the differential equation by the method of exact equation.
ii. Show that the differential equation is homogeneous.
Hence, solve the differential equation by the method of
homogeneous equation. Check the answer with 3i.
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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1.2.4 Linear First Order Differential Equation
How to identify?
The general form of the first order linear DE is given by
When the above equation is divided by
,
(1)
Where
NOTE:
and
Must be
here!!
Method of Solution :
i)
Determine the value of
of
dan
is 1.
ii)
Calculate the integrating factor,
iii)
Write the equation in the form of
iv)
The general solution is given by
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such the the coefficient
CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
Example 1:
Solve this first order DE
Solution:
i)
Determine
ii)
Find integrating factor,
iii)
Write down the equation
iv)
Final answer
and
Note:
Non-linear DE can be converted into linear DE by
using the right substitution.
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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Example 2:
Using
, convert the following non-linear DE into linear DE.
Solve the linear equation.
Solutions:
i)
Differentiate
to get
and replace into the non-
linear equation.
ii)
Change the equation into the general form of linear
equation & determine
and
iii)
Find the integrating factor,
iv)
Find
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
Since
v)
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,
Use the initial condition given,
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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1.2.5 Equations of the form
How to identify?
When the DE is in the form
(1)
use substitution
(2)
to turn the DE into a separable equation
Method of Solution :
i) Differentiate ( 2 ) wrt
( to get
)
(3)
ii) Replace ( 3 ) into ( 1 )
iii) Solve using the separable equation solution
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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Example 1:
Solve
Solution:
i)
Write the equation as a function of
(1)
ii)
Let
and differentiate it to get
(2)
iii)
Replace ( 2 ) into ( 1 )
iv)
Solve using the separable equation solution
Substitution Method

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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
Since
Since
,
,
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1.2.6 Bernoulli Equation
How to identify?
The general form of the Bernoulli equation is given by
(1)
where
To reduce the equation to a linear equation, use substitution
(2)
Method of Solution :
iv) Divide ( 1 ) with
(3)
v) Differentiate ( 2 ) wrt
( to get
)
(4)
vi) Replace ( 4 ) into ( 3 )
vii) Solve using the linear equation solution
 Find integrating factor,
 Solve
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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Example 1:
Solve
(1)
Solutions:
i)
Determine
ii)
Divide ( 1 ) with
(2)
iii)
Using substitution,
(3)
iv)
Replace ( 3 ) into ( 2 ) and write into linear equation form
v)
Find the integrating factor
vi)
Solve the problem
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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vii) Since
Exercise:
Answer:
1.
2.
3.
4.
5.
6. Given the differential equation,
 2x4 y  dy  (4x3 y2  x3)dx  0.
Show that the equation is exact. Hence solve it.
7. The equation in Question 6 can be rewritten as a Bernoulli equation,
2x
dy
 4y  1 .
y
dx
By using the substitution z  y 2 , solve this equation. Check the
answer with Question 6.
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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1.3 Applications of the First Order ODE
The Newton’s Law of Cooling
The Newton’s Law of Cooling is given by the following equation
Where
is a constant of proportionality
is the constant temperature of the surrounding medium
General Solution
Q1: Find the solution for
It is a separable equation. Therefore
Q2: Find
when
Q3: Find . Given that
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Do you know
what type of DE
is this?
CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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Example 1:
According to Newton’s Law of Cooling, the rate of change of the temperature
satisfies
Where is the ambient temperature, is a constant and is time in minutes.
When object is placed in room with temperature
C, it was found that the
temperature of the object drops from 9 C to
C in 30 minutes. Then
determine the temperature of an object after 20 minutes.
Solution:
i) Determine all the information given.
Room temperature =
C
When
When
Question: Temperature after 20 minutes,
ii) Find the solution for
iii) Use the conditions given to find
When
,
and
C
When
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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iv)
Exercise:
1. A pitcher of buttermilk initially at 25⁰C is to be cooled
by setting it on the front porch, where the temperature
is
. Suppose that the temperature of the buttermilk
has dropped to
after 20 minutes. When will it be at
?
2. Just before midday the body of an apparent homicide
victim is found in a room that is kept at a constant
temperature of 70⁰F. At 12 noon, the temperature of the
body is 80⁰F and at 1pm it is 75⁰F. Assume that the
temperature of the body at the time of death was 98.6⁰F
and that it has cooled in accord with Newton’s Law.
What was the time of death?
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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Natural Growth and Decay
The differential equation
Do you know
what type of DE
is this?
Where
is a constant
is the size of population / number of dollars / amount of
radioactive
The problems:
1. Population Growth
2. Compound Interest
3. Radioactive Decay
4. Drug Elimination
Example 1:
A certain city had a population of 25000 in 1960 and a population of
30000 in 1970. Assume that its population will continue to grow
exponentially at a constant rate. What populations can its city
planners expect in the year 2000?
Solution:
1) Extract the information
2) Solve the DE
Separate the equation and integrate
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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3) Use the initial & boundary conditions
In the year 2000, the population size is expected to be
Exercise:
1) (Compounded Interest) Upon the birth of their first child, a couple
deposited RM5000 in an account that pays 8% interest compounded
continuously. The interest payments are allowed to accumulate. How
much will the account contain on the child’s eighteenth birthday? (ANS:
RM21103.48)
2) (Drug elimination) Suppose that sodium pentobarbitol is used to
anesthetize a dog. The dog is anesthetized when its bloodstream contains
at least 45mg of sodium pentobarbitol per kg of the dog’s body weight.
Suppose also that sodium pentobarbitol is eliminated exponentially from
the dog’s bloodstream, with a half-life of 5 hours. What single dose should
be administered in order to anesthetize a 50-kg dog for 1 hour? (ANS: 2585
mg)
3) (Half-life Radioactive Decay) A breeder reactor converts relatively stable
uranium 238 into the isotope plutonium 239. After 15 years, it is
determined that 0.043% of the initial amount
of plutonium has
disintegrated. Find the half-life of this isotope if the rate of disintegration is
proportional to the amount remaining. (ANS: 24180 years)
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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Electric Circuits - RC
Given that the DE for an RL-circuit is
Do you know
what type of DE
is this?
Where
is the voltage source
is the inductance
is the resistance
CASE 1 :
(constant)
(1)
i)
Write in the linear equation form
ii)
Find the integrating factor,
iii)
Multiply the DE with the integrating factor
iv)
Integrate the equation to find
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
CASE 2 :
Consider
i)
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or
, the DE can be written as
Write into the linear equation form and determine
and
ii) Find integrating factor,
iii) Multiply the DE with the integrating factor
iv) Integrate the equation to find
(1)
Tabular Method
Differentiate
Sign
Integrate
+
+
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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(2)
From (1)
(3)
Replace (3) into (2)
Exercise:
1) A 30-volt electromotive force is applied to an LR series circuit in which
the inductance is 0.1 henry and the resistance is 15 ohms. Find the
curve
if
. Determine the current as
.
2) An electromotive force
is applied to an LR series circuit in which the inductance is 20 henries
and the resistance is 2 ohms. Find the current
if
.
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
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Vertical Motion – Newton’s Second Law of Motion
The Newton’s Second Law of Motion is given by
Where
is the external force
is the mass of the body
is the velocity of the body with the same direction with
is the time
Example 1:
A particle moves vertically under the force of gravity against air resistance
where
is a constant. The velocity
at any time is given by the differential
equation
If the particle starts off from rest, show that
Such that
. Then find the velocity as the time approaches
infinity.
Solution:
i)
Extract the information from the question
Initial Condition
ii)
Separate the DE
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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION
Let
iii)
,
Integrate the above equation
Using Partial Fraction
iv)
Use the initial condition,
v)
Rearrange the equation
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vi)
Find the velocity as the time approaches infinity.
When
,
=>
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