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# 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

General Solution Dynamic Equation Prove Theorem Trigonometric Function LAPLACE Transform
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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