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**Example text**

Solution: The strategy to solve this problem is to build an array to represent the x-interval, and then use the difference equation routine to find the value of the Chebyshev polynomial at each value of the array, remembering that the indexing should always be a positive integer. *T(k-1,x1)-T(k-2,x1); end y=T(N,x1); plot(x,y) © 2001 by CRC Press LLC In-Class Exercise Pb. 24 By comparing their plots, verify that the above definition for the Chebyshev polynomial gives the same graph as that obtained from the closed-form expression: TN(x) = cos(N cos–1(x)) for 0 ≤ x ≤ 1 In addition to the Chebyshev polynomials, you will encounter other orthogonal polynomials in your engineering studies.

The following table summarizes the form of the particular solution of a linear equation for some simple input functions: Input Signal Particular Solution A (constant) AMk AkM {A cos(ω0k), A sin(ω0k)} B (constant) BMk B0kM + B1kM–1 + … + BM B1 cos(ω0k) + B2 sin(ω0k) For more complicated input signals, the z-transform technique provides the simplest solution method. This technique is discussed in great detail in courses on linear systems. © 2001 by CRC Press LLC In-Class Exercise Pb. 3 General Solution The general solution of a linear difference equation is the sum of its homogeneous solution and its particular solution, with the constants adjusted, so as to satisfy the initial conditions.

This is a feature not encountered in the classical geometric objects with which you are most familiar. © 2001 by CRC Press LLC In-Class Exercise Pb. 5 Write a program to draw the Koch curve at the kth step. 1, we explored the general numerical techniques for solving difference 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 linear system, the output y(k), k > 0, given a specific input u(k) and a specific set of initial conditions.