MAT128A: Numerical Analysis, Homework 4

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1. Suppose that n is a nonnegative integer. You know that the function y(t) = cos(nt) satisfies the
second order differential equation
y:(t) + n
2
y(t) = 0 for all − π < t < π. Use this observation to show that the function Tn(x) = cos(n arccos(x)) is a solution of the equation (1 − x 2 )y 00(x) − xy0 (x) + n 2 y(x) = 0 for all − 1 < x < 1. Here, I am using y 0 to denote the derivative of y with respect to x and y9 to denote the derivative of y with respect to t. Hint: use the chain rule to compute dy dt and d 2y dt2 in terms of dy dx and d 2y dx2 . 2. Show that Z 1 −1 Tn(x)Tm(x) dx ? 1 − x 2 =    0 m 6= n π m = n = 0 π 2 m = n 6= 0. 3. (a) Using the trigonometric identity cos(nt) = cos((n − 1)t) cos(t) − sin((n − 1)t) sin(t), show that Tn(x) = xTn−1(x) − Un−1(x) a 1 − x 2, (1) where Un is defined via Un(x) = sin(n arccos(x)). (b) Use the trigonometric identity sin(nt) = sin((n − 1)t) cos(t) + cos((n − 1)t) sin(t), to show that Un(x) = Tn−1(x) a 1 − x 2 − Un−1(x)x. (2) 1 (c) Combine (1) and (2) to show that Un(x) a 1 − x 2 = Tn−1(x) + xTn(x). (3) (d) Use (3) and (1) — replace n with n + 1 in (1) — to obtain the recurrence relation Tn+1(x) = 2xTn(x) − Tn−1(x). 4. Suppose that n is a nonnegative integer. Show that (1 − x 2 )T 0 n (x) = nTn−1(x) − nxTn(x) for all −1 < x < 1. 2