Second Order Linear Equations

  • Martin Bohner
  • Allan Peterson

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

Convolution Sino 

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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Martin Bohner
    • 1
  • Allan Peterson
    • 2
  1. 1.Department of MathematicsUniveristy of Missouri-RollaRollaUSA
  2. 2.Department of MathematicsUniversity of NebraskaLincolnUSA

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