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Boundary Value Problems. When the values of a solution to
Chapter 4, Problem 26E(choose chapter or problem)
PROBLEM 26E
Boundary Value Problems. When the values of a solution to a differential equation are specified at two different points, these conditions are called boundary conditions. (In contrast, initial conditions specify the values of a function and its derivative at the same point.) The purpose of this exercise is to show that for boundary value problems there is no existence–uniqueness theorem that is analogous to Theorem 1. Given that every solution to (17) y’’+y = 0 is of the form y(t) = c1cost + c2 sin t, where c1 and c2 are arbitrary constants, show that
(a) There is a unique solution to (17) that satisfies the boundary conditions y(0) = 2 and y(π/2) = 0 .
(b) There is no solution to (17) that satisfies y(0) = 2 and y(π) = 0.
(c) There are infinitely many solutions to (17) that Satisfy y(0) =2 and y(π) = -2.
Questions & Answers
QUESTION:
PROBLEM 26E
Boundary Value Problems. When the values of a solution to a differential equation are specified at two different points, these conditions are called boundary conditions. (In contrast, initial conditions specify the values of a function and its derivative at the same point.) The purpose of this exercise is to show that for boundary value problems there is no existence–uniqueness theorem that is analogous to Theorem 1. Given that every solution to (17) y’’+y = 0 is of the form y(t) = c1cost + c2 sin t, where c1 and c2 are arbitrary constants, show that
(a) There is a unique solution to (17) that satisfies the boundary conditions y(0) = 2 and y(π/2) = 0 .
(b) There is no solution to (17) that satisfies y(0) = 2 and y(π) = 0.
(c) There are infinitely many solutions to (17) that Satisfy y(0) =2 and y(π) = -2.
ANSWER:
SOLUTION
Step 1
Given that every solution to is of the form