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Advance Engineering Maths(213002)
Patel Jaimin -130460119099
Patel Mrugesh-130460119101
Patel Kaushal-130460119105
DATE : 13th November 2014
DIFFERENTIAL
EQUATION
History of the Differential Equation



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Period of the invention
Who invented the idea
Who developed the methods
Background Idea
Differential Equation
Economics
x) 2 y  y  0
yy  (
y  f( x ) dy
dn y
dx n
FUNCTION
y  ex
DERIVATIVE S
2
(-, )
R
dx
 2 xex
2
Chemistry
Mechanics
Engineering
Biology
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,
= f(x,y)
when n=2,
= f(x,y,y1) and so on …
CLASSIFICATIONS BY LINEARITY
Linear
The n th Order ODE is said to be linear if F( x , y , y, y,......, y ( n ) )  0
is linear in y 1 , y 2 , ......., y n
In other words, it has the following general form:
dny
d n 1 y
d2y
dy
a n ( x ) n  a n  1 ( x ) n  1  ......  a2 ( 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
d2y
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
Linear
For Example,
 y  x  dx  5 x dy  0
y  x  5 xy  0
5 xy  y  x
which are linear 1st Order ODE
Likewise,
Linear 2nd Order ODE is
y  5 xy  y  2 x 2
Linear 3rd Order ODE is
y  xy  5 y  e x
Non-Linear
For Example,
1  y  y  5 y  e x
y   cos y  0
y (4)  y 2  0
Classification of Differential Equation
Type:
Ordinary
Partial
Order :
1st, 2nd, 3rd,....,nth
Linearity :
Linear
Non-Linear
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
Problem
In a certain House, a police were called about 3’O Clock where a
murder victim was found.
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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
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was found to be 33.9 C.
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The temperature of the room was 15 C
So, what is the murder time?
“ 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
TIME(t)
TEMPERATURE(ф)
First
t = 0Instant
Ф = 34.5OC
Second
t = 1 Instant
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
 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
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
t
ln
22  ln 19 
 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.