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In-Class Exercise
th
Pb. 2.5 Write a program to draw the Koch curve at the k step. (Hint: Start-
ing with the farthest left vertex and going clockwise, write a difference equa-
tion relating the coordinates of a vertex with those of the preceding vertex,
the length of the segment, and the angle that the line connecting the two con-
secutive vertices makes with the x-axis.)

2.4 Solution of Linear Constant Coefficients Difference
Equations

In Section 2.1, we explored the general numerical techniques for solving dif-
ference equations. In this section, we consider, some special techniques for
obtaining the analytical solutions for the class of linear constant coefficients
difference equations. The related physical problem is to determine, for a lin-
ear system, the output y(k), k > 0, given a specific input u(k) and a specific set
of initial conditions. We discuss, at this stage, the so-called direct method.
The general expression for this class of difference equation is given by:

M
N
∑ ay k( − j) = ∑ b u k( − m) (2.20)
j
m
j=0 m=0
The direct method assumes that the total solution of a linear difference equa-
tion is the sum of two parts — the homogeneous solution and the particular
solution:

y(k) = y homog. (k) + y partic. (k) (2.21)

The homogeneous solution is independent of the input u(k), and the RHS of
the difference equation is equated to zero; that is,

N
∑ ay k( − j) = 0 (2.22)
j
j=0

2.4.1 Homogeneous Solution
Assume that the solution is of the form:

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