Section 4.4

							      Section 4.4

Undetermined Coefficients—
 Superposition Approach
       SOLUTION OF
 NONHOMOGENEOUS EQUATIONS
Theorem: Let y1, y2, . . . , yk be solutions of the
homogeneous linear nth-order differential equation
on an interval I, and let yp be any solution of the
nonhomogeneous equation on the same interval.
Then
   y  c1 y1 ( x)  c2 y2 ( x)   ck yk ( x)  y p ( x)
is also a solution of the nonhomogeneous equation
on the interval for any constants c1, c2, . . . , ck.
 EXISTENCE OF CONSTANTS

Theorem: Let yp be a given solution of the
nonhomogeneous nth-order linear differential
equation on an interval I, and let y1, y2, . . . , yn be
a fundamental set of solutions of the associated
homogeneous equation on the interval. Then for
any solution Y(x) of the nonhomogeneous
equation on I, constants C1, C2, . . . , Cn can be
found so that
    Y  C1 y1 ( x)  C2 y2 ( x)    Cn yn ( x)  y p ( x).
     GENERAL SOLUTION—
  NONHOMOGENEOUS EQUATION
Definition: Let yp be a given solution of the
nonhomogeneous linear nth-order differential equation
on an interval I, and let
        yc  c1 y1 ( x)  c2 y2 ( x)    cn yn ( x)
denote the general solution of the associated
homogeneous equation on the interval. The general
solution of the nonhomogeneous equation on the
interval is defined to be
y  c1 y1 ( x)  c2 y2 ( x)    cn yn ( x)  y p ( x)  yc ( x)  y p ( x).
  COMPLEMENTARY FUNCTION

In the definition from the previous slide, the linear
combination
        yc  c1 y1 ( x)  c2 y2 ( x)    cn yn ( x)
is called the complementary function for the
nonhomogeneous equation. Note that the general
solution of a nonhomogeneous equation is
y = complementary function + any particular solution.
 SUPERPOSITION PRINCIPLE—
NONHOMOGENEOUS EQUATION
Theorem: Let y p , y p ,, y p be k particular solutions of the
                      1     2          k

nonhomogeneous linear nth-order differential equation on an
open interval I corresponding, in turn, to k distinct functions
g1, g2, . . . , gk. That is, suppose y pi denotes a particular
solution of the corresponding DE
       an ( x) y ( n )   a1 ( x) y  a0 ( x) y  g i ( x)
where i = 1, 2, . . . , k. Then
        y p  y p1 ( x)  y p2 ( x)    y pk ( x)

is a particular solution of
               an ( x) y ( n )    a1 ( x) y  a0 ( x) y
                     g1 ( x)  g 2 ( x)   g k ( x).
  FINDING A SOLUTION TO A
NONHOMOGENEOUS LINEAR DE
To find the solution to a nonhomogeneous linear
differential equation with constant coefficients
requires two things:
(i)    Find the complementary function yc.
(ii)   Find any particular solution yp of the
       nonhomoegeneous equation.
LIMITATIONS OF THE METHOD OF
 UNDETERMINED COEFFICIENTS

The method of undetermined coefficients is
limited to nonhomogeneous equations which
• coefficients are constant and
• g(x) is a constant k, a polynomial function,
  an exponential function eαx, sin βx, cos βx, or
  finite sums and products of these functions.
     THE PARTICULAR SOLUTION
       OF ay″ + by′ +cy = g(x)
           g(x)                       yp(x)
Pn(x) = anxn +       xs(Anxn + An−1xn−1 + … + A0)
an−1xn−1 + … + a0
Pn(x)eαx             xs(Anxn + An−1xn−1 + … + A0)eαx
Pn(x)eαx sin βx or   xs [(Anxn + An−1xn−1 + … + A0)eαxcos βx
Pn(x)eαx cos βx      + (Bnxn + Bn−1xn−1 + … + B0)eαxsin βx ]

NOTE: s is the smallest nonnegative integer which
will insure that no term of yp(x) is a solution of the
corresponding homogeneous equation.
HOMEWORK
  1–41 odd