1.  Advance Engineering Maths(213002)  Patel Jaimin -130460119099  Patel Mrugesh-130460119101  Patel Kaushal-130460119105

2. DATE : 13 th November 2014 DIFFERENTIAL EQUATION

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

4. Differential Equation Economics Mechanics Engineering Biology Chemistry

5. 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

6. 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,

7. CLASSIFICATION Differential Equations are classified by : Type,Order,Linearity,

8. 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,

9. 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 …

10. 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

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

12. Classification of Differential Equation  Type: Ordinary Partial  Order : 1 st, 2 nd, 3 rd,....,n th  Linearity : Linear Non-Linear

13. 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

15. 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

16. “ 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

17. 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

18. Mathematically, expression can be written as –

19. 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 = 37 O C 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.