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Ode powerpoint presentation1

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Pokkarn Narkhede
Pokkarn Narkhede

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.

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DATE : 09th October 2012 DIFFERENTIAL EQUATION PRESENTED BY : POKARN NARKHEDE
History of the Differential Equation Period of the invention Who invented the idea Who developed the methods Background Idea
Differential Equation y 緒 ( 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
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 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 n1 2 d y d y d y dy an ( x) n an1( x ) n1 ...... 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
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
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 Bernoullis 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
Ode powerpoint presentation1
Problem In a certain House, a police were called about 3O 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?
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() First0 t = Instant 个 = 34.5OC Second Instant t=1 个 = 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 (个 -15.0) = -0.032t + ln 19 Substituting, 个 = 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.
Ode powerpoint presentation1

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Ode powerpoint presentation1

  • 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 緒 ( 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 n1 2 d y d y d y dy an ( x) n an1( x ) n1 ...... 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 Bernoullis 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 3O 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() First0 t = Instant 个 = 34.5OC Second Instant t=1 个 = 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 (个 -15.0) = -0.032t + ln 19 Substituting, 个 = 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.