Higher Order
    Linear
   Differential
   Equations
   Math 240
 Linear DE
  Linear
  differential
  operators
  Familiar stuff       Higher Order Linear Differential Equations
  Example
 Homogeneous
 equations
                                       Math 240 — Calculus III
                                           Summer 2015, Session II
                                         Tuesday, July 28, 2015
  Higher Order                                                         Agenda
    Linear
  Differential
   Equations
   Math 240
 Linear DE
  Linear
  differential
  operators
  Familiar stuff
  Example     1. Linear differential equations of order n
 Homogeneous       Linear differential operators
 equations
                   Familiar stuff
                   An example
              2. Homogeneous constant-coefficient linear differential
              equations
   Higher Order                                                                                                                Introduction
       Linear
    Differential
     Equations
     Math 240
  Linear DE
   Linear                  Wenowturn our attention to solving linear differential
   differential
   operators
   Familiar stuff           equations of order n. The general form of such an equation is
   Example
  Homogeneous                a (x)y(n) +a (x)y(n−1) +···+a                                               (x)y′ +a (x)y = F(x),
  equations                    0                        1                                         n−1                       n
                           where a ,a ,...,a , and F are functions defined on an
                                           0      1              n
                           interval I.
                           The general strategy is to reformulate the above equation as
                                                                                   Ly =F,
                           where L is an appropriate linear transformation. In fact, L will
                           be a linear differential operator.
  Higher Order                             Linear differential operators
    Linear
  Differential
   Equations
   Math 240   Recall that the mapping D : Ck(I) → Ck−1(I) defined by
 Linear DE    D(f)=f′ is a linear transformation. This D is called the
  Linear      derivative operator. Higher order derivative operators
  differential
  operators   Dk : Ck(I) → C0(I) are defined by composition:
  Familiar stuff
  Example                              k          k−1
 Homogeneous                         D =D◦D ,
 equations    so that
                                                dkf
                                      Dk(f) =       .
                                                dxk
              A linear differential operator of order n is a linear
              combination of derivative operators of order up to n,
                         L=Dn+a Dn−1+···+a                D+a ,
                                     1                n−1       n
              defined by
                       Ly =y(n) +a y(n−1) +···+a          y′ +a y,
                                     1                 n−1      n
              where the a are continous functions of x. L is then a linear
                         i
              transformation L : Cn(I) → C0(I). (Why?)