Advance Engineering Maths(213002)

Patel Jaimin -130460119099 Patel Mrugesh-130460119101 Patel Kaushal-130460119105

Universal College of Engineering & Technology

DATE : 13th November 2014

DIFFERENTIAL EQUATION

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

Differential Equation

)f(

02

xy

yyy

R

),(-

2d

d )(

22

SDERIVATIVE

n

xx

FUNCTION

n

xedx

yey

dx

yxy

Economics

Mechanics

EngineeringBiology

Chemistry

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 LINEARITYLinear

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 yTrignometric Exponential Logarithmic Functions Functions Functions

)()()()( 2,nfor and

)()()( 1,nfor now

)()()()(......)(a)(a

012

2

2

01

012

2

21

1

1nn

xgyxadx

dyxa

dx

ydxa

xgyxadx

dyxa

xgyxadx

dyxa

dx

ydxa

dx

ydx

dx

ydx n

n

n

n

y ......., ,y ,y inlinear is

0),......,,,,F(iflinear be to said is ODEOrder n The

n21

)(th nyyyyx

Linear

For Example,

Likewise,

Linear 2nd Order ODE is

Linear 3rd Order ODE is

Non-Linear

For Example,

ODEOrderlinearxyyx

yxxy

dyxdxxy

1 are which 5

05

0 5

st

xeyyxy

xyyxy

5

25 2

0

0cos

51

2)4(

yy

yy

eyyy x

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 EquationHomogeneous Differential Equation, by Suitable Substitution (i.e. Non-Homogeneous Differential Equation)Exact Differential EquationExact Differential Equation, by Using Integrating FactorLinear Differential EquationLinear Differential Equation, by Suitable SubstitutionBernoulli’s Differential EquationMethod Of Undetermined Co-efficientsMethod Of Reduction of OrderMethod Of Variation of ParametersSolution Of Non-Homogeneous Linear Differential Equation Having nth Order

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

“ 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 Instant

Second Instant

t = 0

t = 1

Ф = 34.5OC

Ф = 33.9OC

1. The temperature of the room 15OC2. The normal body temperature of human being 37OC

Mathematically, expression can be written as –

nintegratio ofconstant theis c'' where

k.t 0.15ln

) ( .... .0.15

alityproportion ofconstant theis k'' where

0.15

0.15

c

FormSeparableVariabledtkd

kdt

ddt

d

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.

minutes 51 hours 3

hours 86.3032.0

19 ln22 ln

t

THANK YOU