Ode powerpoint presentation1

Ode powerpoint presentation1

• 1.
  DATE : 09thOctober 2012 DIFFERENTIAL EQUATION PRESENTED BY : POKARN NARKHEDE
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  History of theDifferential Equation  Period of the invention  Who invented the idea  Who developed the methods  Background Idea
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  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
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  LANGUAGE OF THEDIFFERENTIAL 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
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  DEFINITION A Differential Equationis an equation containing the derivative of one or more dependent variables with respect to one or more independent variables. For example,
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  CLASSIFICATION Differential Equations areclassified by : Type, Order, Linearity,
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  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,
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  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 …
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  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
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  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
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  Classification of DifferentialEquation Type: Ordinary Partial Order : 1st, 2nd, 3rd,....,nth Linearity : Linear Non-Linear
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  METHODS AND TECHNIQUES VariableSeparable 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
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  Problem In a certainHouse, 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?
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  “ The rateof cooling of a body is proportional to the difference between its temperature and the temperature of the surrounding air ” Sir Issac Newton
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  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
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  Mathematically, expression canbe 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
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  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.