Chapter 4.2, Problem 26E

26. 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

$y^{″}+y=0$ $\left(17\right)$

is of the form

$y\left(t\right)=c_1\cos t+c_2\sin t$,

Where $c_1$ and $c_2$ are arbitrary constants, show that

a. There is a unique solution to $\left(17\right)$ that satisfies the boundary conditions $y\left(0\right)=2$ and $y\left(\pi /2\right)=0$.

b. There is no solution to $\left(17\right)$ that satisfies $y\left(0\right)=2$ and $y\left(\pi \right)=0$.

c. There are infinitely many solutions to $\left(17\right)$ that satisfy $y\left(0\right)=2$ and $y\left(\pi \right)=-2$.