Page 112 - Schaum's Outline of Differential Equations

CHAP. 11] THE METHOD OF UNDETERMINED COEFFICIENTS 95

of the form

where A and B are constants to be determined.

Note: (11.3) in its entirety is assumed even when k^ or k 2 is zero, because the derivatives of sines or
cosines involve both sines and cosines.

GENERALIZATIONS

If (f)(x) is the product of terms considered in Cases 1 through 3, take y p to be the product of the corresponding
assumed solutions and algebraically combine arbitrary constants where possible. In particular, if (f)(x) = e^p^x)
is the product of a polynomial with an exponential, assume

where A,- is as in Case 1. If, instead, (f)(x) = e^p^x) sin fix is the product of a polynomial, exponential,
and sine term, or if (f)(x) = e^p^x) cos fix is the product of a polynomial, exponential, and cosine term, then
assume

where - and Bj (j = 0, 1, ..., n) are constants which still must be determined.
A ;
If (f)(x) is the sum (or difference) of terms already considered, then we take y p to be the sum (or difference)
of the corresponding assumed solutions and algebraically combine arbitrary constants where possible.

MODIFICATIONS
If any term of the assumed solution, disregarding multiplicative constants, is also a term of y h (the homoge-
m
neous solution), then the assumed solution must be modified by multiplying it by x , where m is the smallest
m
positive integer such that the product of x with the assumed solution has no terms in common with y h.

LIMITATIONS OF THE METHOD
In general, if (f)(x) is not one of the types of functions considered above, or if the differential equation does
not have constant coefficients, then the method given in Chapter 12 applies.

Solved Problems

2
11.1. Solve /' -y'-2y = 4x .
2
2
x
From Problem 9.1, y h = c^ + c 2e *. Here <j>(x) = 4X , a second-degree polynomial. Using (11.1), we assume that

Thus, and 2. Substituting these results into the differential equation, we have