The document presents information about differential equations including: - A definition of a differential equation as an equation containing the derivative of one or more variables. - Classification of differential equations by type (ordinary vs. partial), order, and linearity. - Methods for solving different types of differential equations such as variable separable form, homogeneous equations, exact equations, and linear equations. - An example problem demonstrating how to use the cooling rate formula to calculate the time of death based on measured body temperatures.

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
y  ( 2 y   ydx  0
y  x)
n
d y
n
Economics
y  f( x )
FUNCTION
2
DERIVATIVE
dy
S
2
y e  2 xe
x x
dx Chemistry
(-  ,  )
R
Mechanics
Biology
Engineering

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
Order ODE is said to be linear if F( x , y , y  , y  ,......, y )  0
th (n)
The n
is linear in y 1 , y 2 , ......., y n
In other words, it has the following general form:
n n1 2
d y d y d y dy
an ( x) n
 an1( x ) n1
 ......  a 2 ( x ) 2
 a1 ( x )  a0 ( x ) y  g( x )
dx dx dx dx
dy
now for n  1, a1 ( x )  a0 ( x ) y  g( x )
dx
2
d y dy
and for n  2, a2 ( x) 2
 a1 ( x )  a0 ( x ) y  g( x )
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 ey ln y
Trignometric Exponential Logarithmic
Functions Functions Functions

10. 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 x y   y  2 x
2
y   x y   5 y  e
x
Linear 3rd Order ODE is
Non-Linear
For Example, 1  y  y  5 y  e
x
y   cos y  0
 y  0
(4) 2
y

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. 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?

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)
First0
t = Instant Ф = 34.5OC
Second Instant
t=1 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 –
d
   15 . 0 
dt
d
 k   15 . 0 
dt
where ' k' is the constant of proportion ality
d
 k .dt .... (Variable Separable Form )
  15 . 0 
ln   15 . 0   k.t  c
where ' c' is the constant of integratio n

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
 ln 22  ln 19 
t    3 . 86 hours
 0 . 032
  3 hours 51 minutes
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.