Advanced Placement Calculus AB 
                   Unit 7   Differential Equations                             
                   Separable Differential Equations 
                    
                    Advanced Placement Calculus AB Course Content 
                    Unit 7  Differential Equations 
                    7.6       Finding General Solutions Using Separation of Variables 
                    7.7       Finding Particular Solutions Using Initial Conditions and Separation of Variables 
                    7.8       Exponential Models with Differential Equations 
                    
                   Separable Differential Equations 
                   Recall that a differential equation is one that contains a derivative. A differential equation of the 
                   form  dy  f x,y  is a separable differential equation if it can be written as a product of a 
                          dx            
                   function of x and a function of y. That is, if  dy  f          x, y  g x h y , then you have a 
                                                                         dx                        
                   separable differential equation.  
                    
                   An alternate form for writing a separable differential equation, and the one that you will use 
                   when asked to find a solution to a separable differential equation, is  
                    
                                              dy  f x,y  g x h y                         dy g x dx 
                                              dx                                     h y          
                                                                                               
                    
                   Examples 
                   Which of the following are separable differential equations? Write each separable differential 
                   equation in the form  dy  g x dx. 
                                             hy  
                                                 
                    
                   (A)       dy 2xy
                             dx
                   (B)       dy  2xy
                             dx
                             dy                        
                   (C)       dx 2cosxy3sin
                   (D)       dy     y
                             dx    x1
                   (E)       dy x y         y
                             dx
                    
                    
                   Advanced Placement Calculus AB 
                   Unit 7   Differential Equations                             
                   Separable Differential Equations 
                    
                   Finding General Solutions Using Separation of Variables 
                   To find a general solution to a differential equation, we use integration. For finding a general 
                   solution to a first-order separable differential equation, integrate both sides of the differential 
                   equation after you have separated the variables. 
                    
                                                  dy g x dx                           dy  g x dx 
                                                                                                   
                                                                                     
                                                h y                                    h y
                                                                                         
                                                                               
                   In order to find a general solution, you will have to be able to find an antiderivative on both sides 
                   of the integral equation. The process is best illustrated with a few examples. 
                    
                   Example 
                   Find the general solution to the differential equation  dy                 2  
                                                                                    dx  xy.
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                   Advanced Placement Calculus AB 
                   Unit 7   Differential Equations                             
                   Separable Differential Equations 
                    
                   Example 
                                                                                    dy    x2
                   Find the general solution to the differential equation  dx  y2 . 
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                    
                   Advanced Placement Calculus AB 
                   Unit 7   Differential Equations                             
                   Separable Differential Equations 
                    
                   Finding Particular Solutions Using Initial Conditions and Separation of Variables 
                   Just like we did when we introduced antiderivatives in Section 4.1, our work began with a 
                   finding general solution to a differential equation. Then, when a specific function value (an 
                   initial condition) was given, we were able to find a particular solution to the differential 
                   equation. The same can be done when separation of variables is required. 
                    
                   Example 
                   Find the particular solution to the differential equation  dy             x  given that  f 0 4. 
                                                                                      dx  y                     