Page 118 - Schaum's Outline of Differential Equations

CHAP. 11] THE METHOD OF UNDETERMINED COEFFICIENTS 101

and the general solution is

Sx
^ f
11.14. Solve y-5y = x e - xe .
x
Sx
2
From Problem 10.3, y h = c^ . Here <j>(x) = x ^ - xe , which is the difference of two terms, each in manageable
2
form. For x ^ we would assume a solution of the form
5x
For xe we would try initially a solution of the form

Sx
But this supposed solution would have, disregarding multiplicative constants, the term e in common with y h. We
are led, therefore, to the modified expression

We now take y p to be the sum of (1) and (2):

Substituting (3) into the differential equation and simplifying, we obtain

Equating coefficients of like terms, we have

from which

Equation (3) then gives

and the general solution is

Supplementary Problems

In Problems 11.15 through 11.26, determine the form of a particular solution to \-(y) = <j>(x) for <j>(x) as given if the solution
3x
2x
to the associated homogeneous equation \-(y) = 0 is y h = Cie + C 2e .