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# Linear ODE Systems Having a Fundamental Matrix of the Form f(Mt)

5-28-2024

## Keywords

Differential Equations, Linear Algebra, Functional Calculus, Dynamic Slope Field, Fundamental Matrix, Non-autonomous Differential Equations, Factorial Function

## Disciplines

Mathematics | Physical Sciences and Mathematics | Science and Mathematics Education

## Abstract

We interweave scaffolded problem statements with exposition and examples to support the reader as they explore specific linear systems of differential equations with variable coefficients, $\vec{x}'(t)=A(t)\vec{x}(t)$ and initial value $\vec{x}(0)$. We begin with a constant $n\times n$ matrix $M$ and a real or complex-valued function $f$, analytic at the eigenvalues of $M$ with $f(0) = 1$, and construct a linear system of differential equations with solutions $x(t)=f(Mt)\vec{x}(0)$, where $t$ is a parameter in some interval including zero. In general, the solutions to the resulting non-autonomous system are more difficult to analyze than solutions to the constant coefficient case. However, some parts of the analysis of the constant coefficient case can be applied in some examples. We then use the system to explore various applications. This approach highlights numerous connections between linear algebra, elementary calculus, functional calculus and differential equations. For example, the coefficient matrix to the newly constructed system can be used to construct a dynamic slope field for the solutions to the initial value problem, as well as linear and quadratic approximations to its solutions formed by $f(Mt)$. To facilitate this approach, we introduce the reader to functional calculus beyond that normally taught in an elementary differential equations course or first course in linear algebra.

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