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CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793 Classification by type - Ordinary Differential Equations (ODE) Contains one or more dependent variables with respect to one independent variable is the dependent variable while is the independent variable is the dependent variable while is the independent variable Dependent Variable: u Independent Variable: t - Partial Differential Equations (PDE) involve one or more dependent variables and two or more independent variables Can you determine which one is the DEPENDENT VARIABLE and which one is the INDEPENDENT VARIABLES from the following equations ??? Dependent Variable: w Independent Variable: x, t Dependent Variable: u Independent Variable: x, y 1 CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793 Classification by order / degree - Order of Differential Equation Determined by the highest derivative - Degree of Differential Equation Exponent of the highest derivative Examples: Order : 1 Degree: 2 a) b) Order : 2 Degree: 1 c) Order : 2 Degree: 1 d) Order : 3 Degree: 4 Classification as linear / nonlinear - Linear Differential Equations Dependent variables and their derivative are of degree 1 Each coefficient depends only on the independent variable A DE is linear if it has the form Examples: 1) 2) 3) 2 CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793 - Nonlinear Differential Equations Dependent variables and their derivatives are not of degree 1 Examples: 1) Order : 1 Degree: 1 2) Order : 1 Degree: 2 3) Order : 3 Degree: 2 Initial & Boundary Value Problems Initial conditions : will be given on specified given point Boundary conditions : will be given on some points Examples : 1) Initial condition 2) Boundary condition Initial Value Problems (IVP) Initial Conditions: Boundary Value Problems (BVP) Boundary Conditions: 3 CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION SSE1793 Solution of a Differential Equation - General Solutions Solution with arbitrary constant depending on the order of the equation - Particular Solutions Solution that satisfies given boundary or initial conditions Examples: (1) Show that the above equation is a solution of the following DE (2) Solutions: (3) (4) Insert (1) and (4) into (2) Proven that is the solution for the given DE. EXERCISE: Show that is the solution of the following DE 4
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