Difference between revisions of "Abel's theorem"
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==Theorem== | ==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 [[ | + | 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 |
$$W(t) = e_{-p+\mu q}(t,t_0)W(t_0)$$ | $$W(t) = e_{-p+\mu q}(t,t_0)W(t_0)$$ | ||
| − | for $t \in \mathbb{T}^{\kappa}$. | + | for $t \in \mathbb{T}^{\kappa}$, where $e_{-p+\mu q}$ denotes the [[delta exponential]]. |
==Proof== | ==Proof== | ||
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.
