Second Order Linear Equations

Martin Bohner and Allan Peterson
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Martin Bohner: Univeristy of Missouri-Rolla, Department of Mathematics
Allan Peterson: University of Nebraska, Department of Mathematics

Chapter Chapter 3 in Dynamic Equations on Time Scales, 2001, pp 81-134 from Springer

Abstract: Abstract In this section we consider the second order linear dynamic equation $$ y^{\Delta \Delta } + p(t)y^\Delta + q(t)y = f(t), $$ where we assume that p, q, f ∊ Crd. If we introduce the operator L2 : C rd 2 → Crd by $$ L_2 y(t) = y^{\Delta \Delta } (t) + p(t)y^\Delta (t) + q(t)y(t) $$ for $$ t \in \mathbb{T}^{\kappa ^2 } $$ , then (3.1) can be rewritten as L2y = f. If y ∊ C rd 2 and L2y(t) - f(t) for all $$ t \in \mathbb{T}^{\kappa ^2 } $$ , then we say y is a solution of L2y = f on T. The fact that L2 is a linear operator (see Theorem 3.1) is why we call equation (3.1) a linear equation. If f(t) = 0 for all $$ t \in \mathbb{T}^{\kappa ^2 } $$ , then we get the homogeneous dynamic equation L2y = 0. Otherwise we say the equation L2y = f is nonhomogeneous. The following principle of superposition is easy to prove and is left as an exercise.

Keywords: General Solution; Dynamic Equation; Prove Theorem; Trigonometric Function; LAPLACE Transform (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1007/978-1-4612-0201-1_3

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