1550078481-Ordinary_Differential_Equations__Roberts_

212 Ordinary Differential Equations

functions on the interval ( -oo, oo), by the representation theorem for n-th

order nonhomogeneous linear differential equations proven in section 4 .1 t here

exists a particular solution, yp(x), of (1) on a ny interval I on which b(x) is

continuous and t he general solution of (1) is y(x) = Yc(x) +yp(x) where Yc(x)

is the complementary solution of the associated homogeneous linear equation

We h ave seen how to solve (2) and find the complementary solut ion, Yc(x),

previously. So the problem of so lving (1) reduces to one of finding the partic-

ular solution yp(x). When the nonhomogeneity b(x) is a polynomial function,

an exponential function, a sine or cosine function, or a finite sum or product

of such functions, then the method of undetermined coefficients may be used

to find a particular solution yp(x). That is, when b(x) is a function which

consists entirely of terms of the form xP, xP e°'x, xP eax sin bx, or xP eax cos bx

where p is a nonnegative integer, a is a constant (perhaps complex), and a

and b are real constants, then the method of undetermined coefficients may

be used. When b(x) is not of this form, then one may try to find a particular

solution by the Laplace transform method which we will discuss in chapter 5.

The Method of Undetermined Coefficients The method of undeter-

mined coefficients for so lving the nonhomogeneous linear differential equation

with constant coefficients (1) consists of (i) solving t he associated homoge-

neous equation (2), (ii) judiciously guessing the form of t he particular so-

lution, Yp(x), of the nonhomogeneous equation (1) with the coefficients left

unspecified- hence, the name undetermined coefficients, (iii) differenti-

ating the assumed form of particular solution n times and substituting the
particular solution and its derivatives into (2), and (iv) determining, if pos-
sible, specific values for t he unspecified coefficients. If all the coefficients are
determined, then we have guessed the correct form of the particular solu-
tion. If some mathematical anomaly occurs along the way, we have guessed
the wrong form of the particular solution and we must guess again or we
have made an error in the calculations. The following three simple examples
illustrate this method.

EXAMPLE 1 Solution of a Nonhomogeneous Linear

Differential Equation with Constant Coefficients

Solve the nonhomogeneous linear differential equation

(3) y" - 3y' + 2y = 5.