to obtain the real solutions
y 3 (n) = rncos nθ
y 4 (n) = rnsin nθ.
The general solution is any linear combination of these functions and can be ex-
pressed
y(n) = rn[c 1 cos nθ +c 2 sin nθ ],
where rand θare defined by equation (10.59) and c 1 and c 2 are arbitrary constants.
Therefore, when complex roots arise, these roots are expressed in polar form in order
to obtain a real solution and imaginary solution to the given difference equation. If
real solutions are desired, then one can take linear combinations of the real solution
and imaginary solution in order to construct a general solution.
Nonhomogeneous Difference Equations
Nonhomogeneous difference equations can be solved in a manner analogous to
the solution of nonhomogeneous differential equations and one may use the method
of undetermined coefficients or the method of variation of parameters to obtain
particular solutions.
Example 10-42. (Undetermined coefficients)
Solve the first order difference equation
L(yn) = yn+1 + 2yn= 3n.
Solution: First solve the homogeneous equation
L(yn) = yn+1 + 2 yn= 0
Assume a solution yn =λn and obtain the characteristic equation λ+ 2 = 0 with
characteristic root λ=− 2. The complementary solution is then yc(n) = c 1 (−2)n,where
c 1 is an arbitrary constant. Next find any particular solution yp(n)which produces
the right-hand side. Analogous to what has been done with differential equations,
examine the differences of the right-hand side of the given equation. Let r(n) = 3n,
then the first difference is a constant since ∆r(n) = r(n+1)−r(n) = 3.The basic terms
occurring in the right-hand side and the difference of the right-hand side are listed
as members of the set S={ 1 , n }. If any member of Soccurs in the complementary
solution, then the set Sis modified by multiplying each term of the set Sby n. If any
