Difference between revisions of "Abel's theorem"
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(Created page with "Let $\mathbb{T}$ be a time scale and consider the dynamic equation defined by the linear operator $$L_2 y(t) = y^{\Delta \Delta}(t) + p(t) y^{\Delta}(t) + q(t)y(t),$$ wher...") |
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| − | Let $\mathbb{T}$ be a time scale and consider the [[dynamic equation]] defined by the linear operator | + | Let $\mathbb{T}$ be a [[time scale]] and consider the [[dynamic equation]] defined by the linear operator |
$$L_2 y(t) = y^{\Delta \Delta}(t) + p(t) y^{\Delta}(t) + q(t)y(t),$$ | $$L_2 y(t) = y^{\Delta \Delta}(t) + p(t) y^{\Delta}(t) + q(t)y(t),$$ | ||
where $p,q$ are [[continuity | rd-continuous]]. | where $p,q$ are [[continuity | rd-continuous]]. | ||
Revision as of 19:22, 5 December 2014
Let $\mathbb{T}$ be a time scale and consider the dynamic equation defined by the linear operator $$L_2 y(t) = y^{\Delta \Delta}(t) + p(t) y^{\Delta}(t) + q(t)y(t),$$ where $p,q$ are rd-continuous.
Theorem: Let $t_0 \in \mathbb{T}^{\kappa}$ and assume $L_2 y = 0$ is regressive. Suppose that $y_1$ and $y_2$ are two solutions of $L_2 y=0$. Then their wronskian satisfies $$W(t) = e_{-p+\mu q}(t,t_0)W(t_0)$$ for $t \in \mathbb{T}^{\kappa}$.
Proof: █
