To prove time delay trajectory for the following figure traces an ellipse for 0 < τ < π 2 Concept Introduction: The general format of the ellipse equation is Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 A, B, C, D, E, F are constants and (x,y) is a set of points. The condition for non-degenerate real ellipse is, B 2 − 4AC < 0 and C | A B 2 D 2 B 2 C E 2 D 2 E 2 F | = C Δ < 0 If C Δ > 0 , ellipse is imaginary, and if Δ = 0 , then ellipse is a point.

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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

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Question

Chapter 12.4, Problem 1E

Interpretation Introduction

Interpretation:

To prove time delay trajectory for the following figure traces an ellipse for 0<τ<π2

Concept Introduction:

The general format of the ellipse equation is

Ax2+ Bxy + Cy2+ Dx + Ey + F = 0

A, B, C, D, E, F are constants and (x,y) is a set of points.

The condition for non-degenerate real ellipse is,

B2−4AC<0 and

C|AB2D2B2CE2D2E2F|=CΔ<0

If CΔ>0, ellipse is imaginary, and if Δ=0, then ellipse is a point.

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