Difference between revisions of "Abel's theorem"

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Let $\mathbb{T}$ be a [[time scale]] and consider the [[dynamic equation]] defined by the linear operator
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==Theorem==
$$L_2 y(t) = y^{\Delta \Delta}(t) + p(t) y^{\Delta}(t) + q(t)y(t),$$
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Let $t_0 \in \mathbb{T}^{\kappa}$ and assume $y^{\Delta \Delta}(t) + p(t) y^{\Delta}(t) + q(t)y(t) = 0$ is regressive, where $p$ and $q$ are [[rd continuous]]. Suppose that $y_1$ and $y_2$ are two solutions of $L_2 y=0$. Then their [[Wronskian]] satisfies
where $p,q$ are [[continuity | rd-continuous]].
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem:</strong> 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
 
 
$$W(t) = e_{-p+\mu q}(t,t_0)W(t_0)$$
 
$$W(t) = e_{-p+\mu q}(t,t_0)W(t_0)$$
for $t \in \mathbb{T}^{\kappa}$.
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for $t \in \mathbb{T}^{\kappa}$, where $e_{-p+\mu q}$ denotes the [[delta exponential]].
<div class="mw-collapsible-content">
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<strong>Proof:</strong> █
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==Proof==
</div>
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</div>
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==References==
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[[Category:Theorem]]
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[[Category:Unproven]]

Latest revision as of 12:48, 16 January 2023

Theorem

Let $t_0 \in \mathbb{T}^{\kappa}$ and assume $y^{\Delta \Delta}(t) + p(t) y^{\Delta}(t) + q(t)y(t) = 0$ is regressive, where $p$ and $q$ are rd continuous. 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 $t \in \mathbb{T}^{\kappa}$, where $e_{-p+\mu q}$ denotes the delta exponential.

Proof

References