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 
                                                                           
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                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)                                                       
                       
                                                 
                                              
                         
                      
                     
                     
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                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: 
                               
                     
                              
                                                                                         
                                                                                      
                 
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                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                  
                                          
                                                   
                 
                 
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