The Homogeneous Linear Equation and Wronskians

Abstract

Abstract The main aim of this chapter will be to use Theorem 4 of Chapter 2 – and specifically both existence and uniqueness of solutions – to develop a theory that will describe the solutions of the homogeneous linear second-order equation where p 0; p 1; p 2 are continuous real-valued functions on [a, b] and p 2(x) < 0 for each x in [a, b]. (‘Homogeneous’ here reflects the zero on the right-hand side of the equation which allows λy to be a solution (for any real constant λ) whenever y is a given solution.) The language of elementary linear algebra will be used and the theory of simultaneous linear equations will be presumed. Central to our discussion will be the Wronskian, or Wronskian determinant: if y 1 : [a, b] → ℝ and y 2 : [a, b] → ℝ are differentiable functions on the closed interval [a, b], the Wronskian of y 1 and y 2, W(y 1; y 2) : [a, b] → ℝ, is defined, for x ∈ [a, b], by If y 1 and y 2 are solutions of (1), it will turn out that either W(y 1, y 2) is identically zero or never zero in [a, b].