November 1997  ICTCM X (Chicago)  2

Past and Present

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.) Given that every solution to y′′ + y = 0 is of the form

y(x) = c_{1} cos x + c_{2} sin x,

where c_{1} and c_{2} are arbitrary constants, show that

(a) there is a unique solution that satisfies the boundary conditions y(0) = 2 and y(π/2) = 0;

(b) there is no solution that satisfies y(0) = 2 and y(π) = 0;

(c) there are infinitely many solutions that satisfy y(0) = 2 and y(π) = −2.

— Fundamentals of Differential Equations, Nagle & Saff (# 30, p. 159)