This is a PowerPoint Presentation on Ordinary Differential Equations.

1. DATE : 09th October 2012
DIFFERENTIAL
EQUATION
PRESENTED BY : POKARN NARKHEDE

2. History of the Differential Equation
 Period of the invention
 Who invented the idea
 Who developed the methods
 Background Idea

3. Differential Equation
)f(
02
xy
yyy


R
),(-
2
d
d
)(
22
SDERIVATIVE
n



xx
FUNCTION
n
xe
dx
y
ey
dx
y
xy 
Economics
Mechanics
Engineering
Biology
Chemistry

4. 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

5. 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,

6. CLASSIFICATION
Differential Equations are classified by : Type, Order, Linearity,

7. 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,

8. 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, = f(x,y)
when n=2, = f(x,y,y1) and so on …

9. CLASSIFICATIONS BY LINEARITY
Linear
In other words, it has the following general form:
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 ey ln y
Trignometric Exponential Logarithmic
Functions Functions Functions
)()()()(2,nforand
)()()(1,nfornow
)()()()(......)(a)(a
012
2
2
01
012
2
21
1
1nn
xgyxa
dx
dy
xa
dx
yd
xa
xgyxa
dx
dy
xa
xgyxa
dx
dy
xa
dx
yd
xa
dx
yd
x
dx
yd
x n
n
n
n


 


y.......,,y,yinlinearis
0),......,,,,F(iflinearbetosaidisODEOrdernThe
n21
)(th
 n
yyyyx

10. Linear
For Example,
Likewise,
Linear 2nd Order ODE is
Linear 3rd Order ODE is
Non-Linear
For Example,
 
ODEOrderlinearxyyx
yxxy
dyxdxxy
1arewhich5
05
05
st



x
eyyxy
xyyxy


5
25 2
 
0
0cos
51
2)4(



yy
yy
eyyy x

11. Classification of Differential Equation
Type: Ordinary Partial
Order : 1st, 2nd, 3rd,....,nth
Linearity : Linear Non-Linear

12. METHODS AND TECHNIQUES
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

14. 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?


Problem


15. “ The rate of cooling of a body is
proportional to the difference
between its temperature and the
temperature of the surrounding
air ”
Sir Issac Newton

16. TIME(t) TEMPERATURE(ф)
First Instant
Second Instant
t = 0
t = 1
Ф = 34.5OC
Ф = 33.9OC
1. The temperature of the room 15OC
2. The normal body temperature of human being 37OC

17. Mathematically, expression can be written as –
 
 
 
 
nintegratioofconstanttheisc''where
k.t0.15ln
)(.....
0.15
alityproportionofconstanttheisk''where
0.15
0.15
c
FormSeparableVariabledtk
d
k
dt
d
dt
d













18. 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
k = ln 18.9 - ln 19
= - 0.032
ln (Ф -15.0) = -0.032t + ln 19
Substituting, Ф = 37OC
ln22 = -0.032t + ln 19
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.
which we gets the murder time.
 
minutes51hours3
hours86.3
032.0
19ln22ln




t