Dynamic equation
Let $\mathbb{T}$ be a time scale. Dynamic equations are time scale analogues of differential equations and difference equations. If we have $\mathbb{T}=\mathbb{R}$ then the resulting theory of dynamic equations is the thoery of differential equations. If $\mathbb{T}=h\mathbb{Z}$ then the resulting theory of dynamic equations is the theory of difference equations of stepsize $h$.
Suppose that $f \colon \mathbb{T} \times \mathbb{R}^2 \rightarrow \mathbb{R}$. Then the equation $$y^{\Delta} = f(t,y,y^{\sigma})$$ is called a first order dynamic equation. Our goal is generally to find all functions $y \colon \mathbb{T} \rightarrow \mathbb{R}$ that satisfies the equation.
Suppose there are rd-continuous functions $p_i \colon \mathbb{T} \rightarrow \mathbb{R}$. Define the operator $$Ly=y^{\Delta^n}+\displaystyle\sum_{k=1}^n p_i y^{\Delta^{n-i}}.$$ We say that the equation $Ly=f$ is an $n$th order dynamic equation.
Examples of Dynamic Equations
- The exponential functions are defined by a first-order linear initial value problem, for regressive $p$: let $s,t \in \mathbb{T}$,
$$y^{\Delta}(t)=p(t)y(t); y(s)=1.$$
- Let $\mathbb{T} \subset (0,\infty)$. The Euler-Cauchy Equations are defined by a second-order linear dynamic equation: let $a,b \in \mathbb{R}$,
$$t \sigma(t)y^{\Delta \Delta}+ aty^{\Delta} + by=0.$$
