Page 111 - Schaum's Outline of Differential Equations

CHAPTER 11

The Method of

Undetermined

Coefficients

The general solution to the linear differential equation L(y) = 0(.v) is given b> Theorem 8.4-as y = y/, + y p
wherey,,denotes one solution to the differential equation and y/, is Ihe general solution to the assoeialed homo-
geneous equation. L(y) = 0. Methods for obtaining y/, when the differential equation has constant coefficients
are given in Chapters 9 and 10. In this chapter and the next, we give methods for obtaining a particular solution
once y/, is known.
y p

SIMPLE FORM OF THE METHOD
The method of undetermined coefficients is applicable onh if <j)(x) and all of its deriuitives can he written
w
in terms of the same finite set of linearK independent functions. hich we denote by {VI(A-), y 2(.v), ... , .V,,(A-)}.
The method is initiated h\ assuming a particular solution of the form

where A]. A 2 ..... A,, denote arhilrar; imiltiplicathe constants. These arhilrar; constants are then evaluated b\
substituting the proposed solution into the given differential equation and equaling the coefficients of like terms.

Case 1. $(x) = p,,(x), an nth-degree polynomial in x. Assume a solution of the form

where Aj-(./ = 0, 1. 2, .... H) is a constant to he determined.

Case 2. $(*! = ke where k and aare known con.slants. Assume a solution of the form
as

where A is a constant to be determined.

Case 3. $(x) = k l sin flx + A; cos fix where fcj, ft;, and ^are known constants. Assume a solution

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