Формула Якобі (реферат)

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Формула Якобі

Нехай $$x_1(t),\text{.}\text{.}\text{.},x_n(t)$$ — лінійно незалежні розв’язки однорідної системи, $$W[x_1,\text{.}\text{.}\text{.},x_n]$$ — визначник Вронського. Обчислимо похідну визначника Вронського.

$$\begin{array}{c}\frac{d}{\text{dt}}W[x_1,\text{.}\text{.}\text{.},x_n]=\frac{d}{\text{dt}}\\ x_{\text{11}}(t)x_{\text{12}}(t)\text{.}\text{.}\text{.}{x}_{1n}(t)\\ x_{\text{21}}(t)x_{\text{22}}(t)\text{.}\text{.}\text{.}{x}_{2n}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x_{n1}(t)x_{n2}(t)\text{.}\text{.}\text{.}{x}_{\text{nn}}(t)\\ \\ \mathit{rli}\\ \\ =\\ \\ \\ \\ =\frac{d}{\text{dt}}\\ \\ x_{\text{11}}(t)x_{\text{21}}(t)\text{.}\text{.}\text{.}{x}_{n1}(t)\\ x_{\text{12}}(t)x_{\text{22}}(t)\text{.}\text{.}\text{.}{x}_{n2}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x_{1n}(t)x_{2n}(t)\text{.}\text{.}\text{.}{x}_{\text{nn}}(t)\\ \\ \\ \begin{array}{c}||||||\end{array}\\ \\ \begin{array}{c}\end{array}\\ \hfill \end{array}$$.

$$\begin{array}{c}x{\text{'}}_{\text{11}}(t)x_{\text{21}}(t)\text{.}\text{.}\text{.}{x}_{n1}(t)\\ x{\text{'}}_{\text{12}}(t)x_{\text{22}}(t)\text{.}\text{.}\text{.}{x}_{n2}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x{\text{'}}_{1n}(t)x_{2n}(t)\text{.}\text{.}\text{.}{x}_{\text{nn}}(t)\\ \\ \mathit{rli}\\ \\ x_{\text{11}}(t)x{\text{'}}_{\text{21}}(t)\text{.}\text{.}\text{.}{x}_{n1}(t)\\ x_{\text{12}}(t)x{\text{'}}_{\text{22}}(t)\text{.}\text{.}\text{.}{x}_{n2}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x_{1n}(t)x{\text{'}}_{2n}(t)\text{.}\text{.}\text{.}{x}_{\text{nn}}(t)\\ \\ \mathit{rli}\\ \\ +\text{.}\text{.}\text{.}+\\ \\ \\ \\ x_{\text{11}}(t)x_{\text{21}}(t)\text{.}\text{.}\text{.}x{\text{'}}_{n1}(t)\\ x_{\text{12}}(t)x_{\text{22}}(t)\text{.}\text{.}\text{.}x{\text{'}}_{n2}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x_{1n}(t)x_{2n}(t)\text{.}\text{.}\text{.}x{\text{'}}_{\text{nn}}(t)\\ \\ \mathit{rli}\\ \\ \\ \begin{array}{c}||||||\end{array}\\ =\\ \\ \begin{array}{c}\end{array}\\ \hfill \end{array}$$.

Оскільки для похідних виконується співвідношення.

$$\begin{array}{c}x{\text{'}}_{\text{11}}(t)={}_{\text{11}}{x}_{\text{11}}(t)+{}_{\text{12}}{x}_{\text{12}}(t)+\text{.}\text{.}\text{.}+{}_{1n}{x}_{1n}(t)\\ x{\text{'}}_{\text{12}}(t)={}_{\text{11}}{x}_{\text{21}}(t)+{}_{\text{12}}{x}_{\text{22}}(t)+\text{.}\text{.}\text{.}+{}_{1n}{x}_{2n}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x{\text{'}}_{1n}(t)={}_{\text{11}}{x}_{n1}(t)+{}_{\text{12}}{x}_{n2}(t)+\text{.}\text{.}\text{.}+{}_{1n}{x}_{\text{nn}}(t),\\ \\ \begin{array}{c}\{\{\{\end{array}\hfill \\ \end{array}$$.

$$\begin{array}{c}x{\text{'}}_{\text{21}}(t)={}_{\text{21}}{x}_{\text{11}}(t)+{}_{\text{22}}{x}_{\text{21}}(t)+\text{.}\text{.}\text{.}+{}_{2n}{x}_{n1}(t)\\ x{\text{'}}_{\text{22}}(t)={}_{\text{21}}{x}_{\text{12}}(t)+{}_{\text{22}}{x}_{\text{22}}(t)+\text{.}\text{.}\text{.}+{}_{2n}{x}_{n2}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x{\text{'}}_{2n}(t)={}_{\text{21}}{x}_{1n}(t)+{}_{\text{22}}{x}_{2n}(t)+\text{.}\text{.}\text{.}+{}_{2n}{x}_{\text{nn}}(t),\\ \\ \begin{array}{c}\{\{\{\end{array}\hfill \\ \end{array}$$.

…

$$\begin{array}{c}x{\text{'}}_{n1}(t)={}_{n1}{x}_{\text{11}}(t)+{}_{n2}{x}_{\text{21}}(t)+\text{.}\text{.}\text{.}+{}_{\text{nn}}{x}_{n1}(t)\\ x{\text{'}}_{n2}(t)={}_{n1}{x}_{\text{12}}(t)+{}_{n2}{x}_{\text{22}}(t)+\text{.}\text{.}\text{.}+{}_{\text{nn}}{x}_{n2}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x{\text{'}}_{\text{nn}}(t)={}_{n1}{x}_{1n}(t)+{}_{n2}{x}_{2n}(t)+\text{.}\text{.}\text{.}+{}_{\text{nn}}{x}_{\text{nn}}(t),\\ \\ \begin{array}{c}\{\{\{\end{array}\hfill \\ \end{array}$$.

то після підстановки одержимо.

$$\begin{array}{c}{}_{\text{11}}{x}_{\text{11}}(t)+{}_{\text{12}}{x}_{\text{12}}(t)+\text{.}\text{.}\text{.}+{}_{1n}{x}_{1n}(t)x_{\text{21}}(t)\text{.}\text{.}\text{.}{x}_{n1}(t)\\ {}_{\text{11}}{x}_{\text{21}}(t)+{}_{\text{12}}{x}_{\text{22}}(t)+\text{.}\text{.}\text{.}+{}_{1n}{x}_{2n}(t)x_{\text{22}}(t)\text{.}\text{.}\text{.}{x}_{n2}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ {}_{\text{11}}{x}_{n1}(t)+{}_{\text{12}}{x}_{n2}(t)+\text{.}\text{.}\text{.}+{}_{1n}{x}_{\text{nn}}(t)x_{2n}(t)\text{.}\text{.}\text{.}{x}_{\text{nn}}(t)\\ \\ \mathit{rli}\\ \\ \\ \begin{array}{c}||||||\end{array}\\ \frac{d}{\text{dt}}W[x_1,\text{.}\text{.}\text{.},x_n]=\\ \end{array}$$ $$\begin{array}{c}{x}_{\text{11}}(t){}_{\text{21}}{x}_{\text{11}}(t)+{}_{\text{22}}{x}_{\text{21}}(t)+\text{.}\text{.}\text{.}+{}_{2n}{x}_{n1}(t)\text{.}\text{.}\text{.}{x}_{n1}(t)\\ x_{\text{12}}(t){}_{\text{21}}{x}_{\text{12}}(t)+{}_{\text{22}}{x}_{\text{22}}(t)+\text{.}\text{.}\text{.}+{}_{2n}{x}_{n2}(t)\text{.}\text{.}\text{.}{x}_{n2}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x_{1n}(t){}_{\text{21}}{x}_{1n}(t)+{}_{\text{22}}{x}_{2n}(t)+\text{.}\text{.}\text{.}+{}_{2n}{x}_{\text{nn}}(t)\text{.}\text{.}\text{.}{x}_{\text{nn}}(t)\\ \\ \mathit{rli}\\ \\ \\ \begin{array}{c}||||||\end{array}\\ +\\ \end{array}$$.

$$\begin{array}{c}{x}_{\text{11}}(t)\text{.}\text{.}\text{.}{x}_{n-11}(t){}_{n1}{x}_{\text{11}}(t)+{}_{n2}{x}_{\text{21}}(t)+\text{.}\text{.}\text{.}+{}_{\text{nn}}{x}_{n1}(t)\\ x_{\text{12}}(t)\text{.}\text{.}\text{.}{x}_{n-12}(t){}_{n1}{x}_{\text{12}}(t)+{}_{n2}{x}_{\text{22}}(t)+\text{.}\text{.}\text{.}+{}_{\text{nn}}{x}_{n2}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x_{1n}(t)\text{.}\text{.}\text{.}{x}_{n-1n}(t){}_{n1}{x}_{1n}(t)+{}_{n2}{x}_{2n}(t)+\text{.}\text{.}\text{.}+{}_{\text{nn}}{x}_{\text{nn}}(t)\text{.}\\ \\ \mathit{rli}\\ \\ \\ \begin{array}{c}||||||\end{array}\\ +\\ \end{array}$$.

Розкривши кожний з визначників, і з огляду на те, що визначники з однаковими стовпцями дорівнюють нулю, одержимо.

$$\begin{array}{c}\frac{d}{\text{dt}}W[x_1,\text{.}\text{.}\text{.},x_n]=a_{\text{11}}\\ x_{\text{11}}(t)x_{\text{21}}(t)\text{.}\text{.}\text{.}{x}_{n1}(t)\\ x_{\text{12}}(t)x_{\text{22}}(t)\text{.}\text{.}\text{.}{x}_{n2}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x_{1n}(t)x_{2n}(t)\text{.}\text{.}\text{.}{x}_{\text{nn}}(t)\\ \\ \mathit{rli}\\ \\ \\ \begin{array}{c}||||||\end{array}\\ \end{array}$$.

$$\begin{array}{c}+a_{\text{22}}\\ x_{\text{11}}(t)x_{\text{21}}(t)\text{.}\text{.}\text{.}{x}_{n1}(t)\\ x_{\text{12}}(t)x_{\text{22}}(t)\text{.}\text{.}\text{.}{x}_{n2}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x_{1n}(t)x_{2n}(t)\text{.}\text{.}\text{.}{x}_{\text{nn}}(t)\\ \\ \mathit{rli}\\ \\ +\text{.}\text{.}\text{.}+a_{\text{nn}}\\ \\ x_{\text{11}}(t)x_{\text{21}}(t)\text{.}\text{.}\text{.}{x}_{n1}(t)\\ x_{\text{12}}(t)x_{\text{22}}(t)\text{.}\text{.}\text{.}{x}_{n2}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x_{1n}(t)x_{2n}(t)\text{.}\text{.}\text{.}{x}_{\text{nn}}(t)\\ \\ \mathit{rli}\\ \\ \\ \begin{array}{c}||||||\end{array}\\ \end{array}$$.

$$\begin{array}{c}=(a_{\text{11}}+a_{\text{22}}+\text{.}\text{.}\text{.}+a_{\text{nn}})\\ x_{\text{11}}(t)x_{\text{21}}(t)\text{.}\text{.}\text{.}{x}_{n1}(t)\\ x_{\text{12}}(t)x_{\text{22}}(t)\text{.}\text{.}\text{.}{x}_{n2}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x_{1n}(t)x_{2n}(t)\text{.}\text{.}\text{.}{x}_{\text{nn}}(t)\\ \\ \mathit{rli}\\ \\ \\ \begin{array}{c}||||||\end{array}\\ \end{array}$$.

$$\begin{array}{c}{x}_{\text{11}}(t)x_{\text{21}}(t)\text{.}\text{.}\text{.}{x}_{n1}(t)\\ x_{\text{12}}(t)x_{\text{22}}(t)\text{.}\text{.}\text{.}{x}_{n2}(t)\\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\\ x_{1n}(t)x_{2n}(t)\text{.}\text{.}\text{.}{x}_{\text{nn}}(t)\\ \\ \mathit{rli}\\ \\ \\ \begin{array}{c}||||||\end{array}\\ =\text{SpA}\\ \end{array}$$ $$\text{SpA}=\underset{i=1}{\overset{n}{}}{a}_{\text{ii}}$$ .

Або.

$$\frac{d}{\text{dt}}W[x_1,\text{.}\text{.}\text{.},x_n]=\text{SpA}W[x_1,\text{.}\text{.}\text{.},x_n]$$ .

Розділивши змінні, одержимо.

$$\frac{\text{dW}[x_1,\text{.}\text{.}\text{.},x_n]}{W[x_1,\text{.}\text{.}\text{.},x_n]}=\text{SpA}\text{dt}$$ .

Проінтегруємо в межах $$t_0<=s<=t$$ ,.

$$\text{ln}W[x_1(t),\text{.}\text{.}\text{.},x_n(t)]-\text{ln}W[x_1(t_0),\text{.}\text{.}\text{.},x_n(t_0)]={}_{t_0}^t\text{SpAdt}$$ ,.

або.

$$W[x_1(t),\text{.}\text{.}\text{.},x_n(t)]=W[x_1(t_0),\text{.}\text{.}\text{.},x_n(t_0)]e^{{}_{t_0}^t\text{SpAdt}}$$ .

Взагалі кажучи, доведення проводилося в припущенні, що система рівнянь може залежати від часу, тобто.

$$W[x_1(t),\text{.}\text{.}\text{.},x_n(t)]=W[x_1(t_0),\text{.}\text{.}\text{.},x_n(t_0)]e^{{}_{t_0}^t\text{SpA}(t)\text{dt}}$$ .

Отримана формула називається формулою Якобі.