INTRODUCTION TO
DIFFERENTIAL
EQUATION
PRESENTED BY : ANASTACIO G. PANTALEON, JR.
WIDE APPLICATIONS OF
DIFFERENTIAL EQUATION
Economics
Mechanics
Physics
Engineering

Chemistry
Biology
Crime Analytics
Social Science
LANGUAGE OF THE DIFFERENTIAL EQUATION
DEGREE OF ODE
➢ ORDER OF ODE
➢ SOLUTIONS OF ODE
❖ GENERAL SOLUTION
❖ PARTICULAR SOLUTION
❖ TRIVIAL SOLUTION
❖ SINGULAR SOLUTION
❖ EXPLICIT AND IMPLICIT SOLUTION
➢ HOMOGENEOUS EQUATIONS
➢ NON-HOMOGENEOUS EQUTIONS
➢ INTEGRATING FACTOR
➢
DEFINITION
A Differential Equation is an equation containing the derivative of one or
more dependent variables with respect to one or more independent
variables.
For example,
CLASSIFICATION
Differential Equations are classified by : Type, Order, Linearity,
Classifiation by Type:
Ordinary Differential Equation
If a Differential Equations contains only ordinary derivatives of one or
more dependent variables with respect to a single independent variables, it
is said to be an Ordinary Differential Equation or (ODE) for short.
For Example,
Partial Differential Equation
If a Differential Equations contains partial derivatives of one or more
dependent variables of two or more independent variables, it is said to be a
Partial Differential Equation or (PDE) for short.
For Example,
Classifiation by Order:
The order of the differential equation (either ODE or PDE) is the order of the
highest derivative in the equation.
For Example,
Order = 3
Order = 2
Order = 1
General form of nth Order ODE is
= f(x,y,y1,y2,….,y(n))
where
f is a real valued continuous function.
This is also referred to as Normal Form Of nth Order Derivative
So,
when n=1,
when n=2,
= f(x,y)
= f(x,y,y1) and so on …
CLASSIFICATIONS BY LINEARITY
Linear
(n)
The n th Order ODE is said to be linear if F( x , y , y  , y  ,......, y ) = 0
is linear in y 1 , y 2 , ......., y n
In other words, it has the following general form:
dny
an ( x)
dx
n
+ an−1( x )
now for n = 1,
and for n = 2,
d n−1 y
dx
n−1
+ ...... + a 2 ( x )
d2y
dx
2
+ a 1 ( x ) dy + a 0 ( x ) y = g ( x )
dx
a 1 ( x ) dy + a 0 ( x ) y = g ( x )
dx
d2y
+ a 1 ( x ) dy + a 0 ( x ) y = g ( x )
a2( x)
2
dx
dx
Non-Linear :
A nonlinear ODE is simply one that is not linear. It contains nonlinear
functions of one of the dependent variable or its derivatives such as:
siny
ln y
ey
Trignometric
Exponential
Logarithmic
Functions
Functions
Functions
LINEAR
For Example,
( y − x ) dx
+ 5 x dy = 0
y − x + 5 xy = 0
5 xy + y = x
st
which are linear 1 Order ODE
Likewise,
Linear 2nd Order ODE
is
y  − 5 xy + y = 2 x
Linear 3rd Order ODE
is
y  + xy − 5 y = e
2
x
Non-Linear
For Example,
(1
− y
)y
+ 5 y = e
y  + cos y = 0
y
(4 )
+ y
2
= 0
x
SUMMARY ON THE
CLASSIFICATION OF DIFFERENTIAL
EQUATION
➢Type:
Ordinary
Partial
➢Order :
1st, 2nd, 3rd,....,nth
➢Linearity :
Linear
Non-Linear
METHODS AND TECHNIQUES COVERED BY OUR
SUBJECT
• ➢VARIABLE SEPARABLE FORM
• ➢VARIABLE SEPARABLE FORM, BY SUITABLE SUBSTITUTION
• ➢HOMOGENEOUS DIFFERENTIAL EQUATION
• ➢HOMOGENEOUS DIFFERENTIAL EQUATION, BY SUITABLE
SUBSTITUTION
(I.E. NON-HOMOGENEOUS DIFFERENTIAL EQUATION)
➢EXACT DIFFERENTIAL EQUATION
➢EXACT DIFFERENTIAL EQUATION, BY USING INTEGRATING
FACTOR
➢LINEAR DIFFERENTIAL EQUATION
➢LINEAR DIFFERENTIAL EQUATION, BY SUITABLE
SUBSTITUTION
➢BERNOULLI’S DIFFERENTIAL EQUATION
➢METHOD OF UNDETERMINED CO-EFFICIENTS
➢METHOD OF REDUCTION OF ORDER
➢METHOD OF VARIATION OF PARAMETERS
➢SOLUTION OF NON-HOMOGENEOUS LINEAR DIFFERENTIAL
EQUATION HAVING NTH
ORDER
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PROBLEM
In a certain House, a police were called about 3’O Clock where a
murder victim was found.

Police took the temperature of body which was found to be34.5 C.
After 1 hour, Police again took the temperature of the body which

was found to be 33.9 C.

The temperature of the room was 15 C
So, what is the murder time?
Sir Issac Newton
“ THE RATE OF
COOLING OF A BODY IS
PROPORTIONAL TO
THE DIFFERENCE
BETWEEN ITS
TEMPERATURE AND
THE TEMPERATURE OF
THE SURROUNDING AIR
”
TEMPERATURE(ф)
TIME(T)
First Constant
Ф = 34.5OC
Second Constant
Ф = 33.9OC
1. The temperature of the room 15OC
2. The normal body temperature of human being
37OC
MATHEMATICALLY, EXPRESSION CAN BE WRITTEN AS –
d
 ( − 15 .0 )
dt
d
dt
= k ( − 15 .0 )
where ' k' is the constant
d
= k .dt
( − 15 .0 )
ln ( − 15 .0 ) = k.t
where
' c' is the
of proportion
.... (Variable
ality
Separable
+c
constant
of
integration
Form )
ln (34.5 -15.0) = k(0) + c
c = ln19.5
ln (33.9 -15.0) = k(1) + c
ln 18.9 = k+ ln 19.5
k = ln 18.9 - ln 19.5
= - 0.032
= -0.032t + ln 19.5
ln (Ф -15.0)
Substituting, Ф = 37OC
ln22 = -0.032t + ln 19.5
t
=
( ln
2 2 − ln 1 9 . 5 )
− 0 .032
= − 3 . 8 6 h ou r s
= − 3 h ou r s 5 1 m in u te s
So, subtracting the time four our zero instant of time
i.e., 3:45 a.m. – 3hours 51 minutes
i.e., 11:54 p.m.