To explain mathematically the result for x n + 1 = x n using a cobweb or some other argument. Also, use the pocket calculator to explore the maps. Concept Introduction: ➢ Determine the fixed points for the given equation and its stability. ➢ Plot the graph for the equation and compute numerical values.

Question

4. Consider Chebychev's equation (1 - x²)y" - xy + λy = 0 with boundary conditions y(-1) = 0 and y(1) = 0, where X is a constant. (a) Show that Chebychev's equation can be expressed in Sturm-Liouville form d · (py') + qy + Ary = 0, dx y(1) = 0, y(-1) = 0, where p(x) = (1 = x²) 1/2, q(x) = 0 and r(x) = (1 − x²)-1/2 (b) Show that the eigenfunctions of the Sturm-Liouville equation are extremals of the functional A[y], where A[y] = I[y] J[y]' and I[y] and [y] are defined by - I [y] = √, (my² — qy²) dx and J[y] = [[", ry² dx. Explain briefly how to use this to obtain estimates of the smallest eigenvalue >1. 1 (c) Let k > be a parameter. Explain why the functions y(x) = (1-x²) are suitable 4 trial functions for estimating the smallest eigenvalue. Show that the value of A[y] for these trial functions is 4k2 A[y] = = 4k - 1' and use this to estimate the smallest eigenvalue \1. Hint: L₁ x²(1 − ²)³¹ dr = 1 (1 - x²)³ dx (ẞ > 0). 2ẞ

Answer 1

Question

Chapter 10.1, Problem 1E

Interpretation Introduction

Interpretation:

To explain mathematically the result for xn+1 = xn using a cobweb or some other argument. Also, use the pocket calculator to explore the maps.

Concept Introduction:

➢ Determine the fixed points for the given equation and its stability.

➢ Plot the graph for the equation and compute numerical values.

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