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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 ...

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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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...Chapter first order ordinary differential equation sse classification by type equations ode contains one or more dependent variables with respect to independent variable is the while u t partial pde involve and two can you determine which from following w x y degree of determined highest derivative exponent examples a b c d as linear nonlinear their are each coefficient depends only on de if it has form derivatives not initial boundary value problems conditions will be given specified point some points condition ivp bvp solution general solutions arbitrary constant depending particular that satisfies...

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