Q2*) Consider the extremisation of the integral I[y] = √²² F(x,y,y', y") dx x1 when y and y' are prescribed only at x = x1. Derive the so-called 'natural boundary conditions' that must be satisfied at x = x2. Taking a specific example: The functional I [y] is defined by I[y] = √² ((y″)² + y) dx with y(0) = 0 and y'(0) = 0. Write down the fourth-order Euler-Lagrange equation for this problem, stating the four boundary conditions. Find the general solution of the Euler-Lagrange equation, and then impose the boundary conditions to find the extremal.

Q2*) Consider the extremisation of the integral I[y] = √²² F(x,y,y', y") dx x1 when y and y' are prescribed only at x = x1. Derive the so-called 'natural boundary conditions' that must be satisfied at x = x2. Taking a specific example: The functional I [y] is defined by I[y] = √² ((y″)² + y) dx with y(0) = 0 and y'(0) = 0. Write down the fourth-order Euler-Lagrange equation for this problem, stating the four boundary conditions. Find the general solution of the Euler-Lagrange equation, and then impose the boundary conditions to find the extremal.

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

ChapterA: Appendix

SectionA.2: Geometric Constructions

Problem 10P: A soda can has a volume of 25 cubic inches. Let x denote its radius and h its height, both in...

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