HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS 
            
           1     Higher−Order Differential Equations  
                         
                                                             (n)         (n-1)
                        Consider the differential equation: y  + p   (x) y   + . . . + p (x) y' + p (x) y   =   0 
                                                                  n 1                 1        0
                                                                   −
                                                                                    
                 General Solution 
                                                          th
                        A general solution of the above n  order homogeneous linear differential equation on some interval I is a function of the form 
                                                                       y(x)   =   c  y (x) + c  y (x) + ... + c  y (x) 
                                                                                1  1      2 2           n n
                        where y1 , . . ., yn are linearly independent solutions (basis) on I. 
                  
                  
                 Theorem − Existence and Uniqueness of IVP 
                        If the p0(x), p1(x), . . ., pn 1(x) in the differential equations are continuous on an open interval I, then the initial value problem [with 
                                               −
                        x0  in I] has a unique solution in I. 
                  
                  
                                                                                                                                   Higher-Order ODE - 1 
                               Wronskian 
                                           The Wronskian of y , y , . . ., y  is defined as 
                                                                            1    2            n
                                                                                             y1                  y2         ...       yn       
                                                                                             y1'                 y2'        ...       yn'      
                                                                                             y1''               y2''        ...      yn''      
                                                       W(y , y , . . ., y )   =                                                                        
                                                              1     2           n                                                              
                                                                                             ...                  ...       ...       ...      
                                                                                                       −             −                    −
                                                                                            y1(n 1)           y2(n 1)       ...    yn(n 1) 
                                
                               Theorem − Linear Dependence and Independence of Solutions  
                                
                                           Let p (x), p (x), . . ., p             (x)  be continuous in I, [x , x ], and let y , y , . . ., y  be n solutions of the differential equation.  Then 
                                                   0         1                n 1                                          0     1                 1     2           n
                                                                               −
                                           (1)         W(y , y , . . ., y ) is either zero for all x ∈  I or for no value of x ∈  I. 
                                                              1    2           n
                                           (2)         y , y , . . ., y  are linearly independent if and only if 
                                                         1    2           n
                                                                               W(y , y , . . ., y ) ≠  0 
                                                                                      1    2           n
                                
                               Theorem − Existence of a General Solution  
                                
                               Theorem − General Solution  
                                                                                                                                                                                                                                      Higher-Order ODE - 2 
                       
                      [Exercise]  Consider the third−order equation 
                                                y''' + a(x) y'' + b(x) y' + c(x) y   =   0 
                               where a, b and c are continuous functions of x in some interval I.  The Wronskian of y (x), y (x), and y (x) is defined as 
                                                                                                                                           1       2            3
                                                             y1   y2    y3
                                                                             
                                                                             
                                                W   =      y1'    y2'   y3'       
                                                                             
                                                           y1'' y2'' y3'' 
                               where y , y  and y  are solutions of the differential equation.   
                                         1   2        3
                               (a)     Show that W satisfies the differential equation W' + a(x) W   =   0 
                               (b)     Prove that W is always zero or never zero. 
                                                                                       th
                               (c)     Can you extend the above results to n –order linear differential equations? 
                                                                                                                                                                      Higher-Order ODE - 3 
                                          2       nth-Order Homogeneous Equations with Constant Coefficients 
                                                                                                      
                                                       (n)         (n-1)
                                                     y  + a       y     + . . . + a  y' + a  y   =   0                             Differential Equation 
                                                              n 1                  1       0
                                                               −
                                                    λn + a     λn-1 + . . . + a  λ + a    =   0                                  Characteristic Equation 
                                                           n 1                1       0
                                                            −
                                                                                                          
                                                                            Case I           Distinct Roots, λ , λ , . . ., λ  
                                                                                                                       1   2         n
                                                                           The corresponding linearly independent solutions are 
                                                                                                    λ x   λ x        λ x
                                                                                                  e 1  , e 2  , . . ., e n   
                                                                                                          
                                                               Case II          Multiple Roots, λ    =   λ    =   . . .   =   λ    =   λ 
                                                                                                          1         2                    m
                                                                           The corresponding linearly independent solutions are 
                                                                                            λx      λx    2 λx          m−1 λx
                                                                                                            
                                                                                           e   , x e    , x e   , . . ., x  e     
                                                                                                          
                                                        Case III         Complex Simple Roots    λ    =   γ + i ω   ,    λ    =   γ − i ω 
                                                                                                                1                         2
                                                                           The corresponding linearly independent solutions are 
                                                                                               eγx  cos ωx   ,   eγx  sin ωx 
                                                                                                          
                                                                                                                                                                     Higher-Order ODE - 4