Assignment 10 Math427K-H Due Wednesday April 21, 2010 1. Find the recursion relation for the coefficients of the series solution about 0 to the equation: (1-x^2)y" + xy' + y = 0. 2. Find a_j, j = 2,3,4,5,6 if y(0) =1 and y'(0)= 0. 3. Consider Fourier series on the interval [-pi,pi]] and assume the functions below are defined on this interval. Compute the Fourier series for |x| on this interval. Find the Fourier series for f(x) = H(x) (the Heavyside, sgn or square wave function on this same interval)in the text. How are they related? Note: sgn(x) means the sign of x, +1 for x> 0 and -1 for x < 0. Does it matter what x(0)is? 4. Using your answer from #3 , compute the cosine series of 1 - |x| on the interval [0,pi] and the sine series for f(x) = 2 on the same interval. Explain how the series are related. 5. Suppose you have computed the sine series for the function f(x) = x^2 on the interval [0,4]. a) Give the general form of the series (it is not necessary to compute the coefficients in the series.) b) What number does this series sum to at x = - 2? c) What number does this series sum to at x = 4? d) What number does this series sum to at x = 6? Extra Credit: Due Thursday April 22. 1. Compute the Fourier series of the function (sin(x))^3 on the interval [-\pi, \pi]. (Hint: try your hand at trig identities).