 Advance Engineering Maths(213002)  Patel Jaimin  Patel Mrugesh  Patel Kaushal

DATE : 13 th November 2014 DIFFERENTIAL EQUATION

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

Differential Equation Economics Mechanics Engineering Biology 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,y 1,y 2,….,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,y 1 ) and so on …

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 e y ln y Trignometric Exponential Logarithmic Functions Functions Functions

Linear For Example, Likewise, Linear 2 nd Order ODE is Linear 3 rd Order ODE is Non-Linear For Example,

Classification of Differential Equation  Type: Ordinary Partial  Order : 1 st, 2 nd, 3 rd,....,n th  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 n th 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 = 34.5 O C Ф = 33.9 O C 1.The temperature of the room 15 O C 2. The normal body temperature of human being 37 O C

Mathematically, expression can be written as –

ln ( ) = k(0) + c c = ln19.5 ln ( ) = k(1) + c ln 18.9 = k+ ln 19 k = ln ln 19 = ln (Ф -15.0) = t + ln 19 Substituting, Ф = 37 O C ln22 = t + 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.