From 76571427d2b5ecd384acfe576480b24dc1451b23 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Sat, 17 May 2025 21:51:37 +0200 Subject: [PATCH] nazvy vet + chybejici zavorky --- linearni-rovnice.tex | 6 +++--- skripta.pdf | Bin 227164 -> 227256 bytes 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/linearni-rovnice.tex b/linearni-rovnice.tex index 54f6fa8..bc4106e 100644 --- a/linearni-rovnice.tex +++ b/linearni-rovnice.tex @@ -46,7 +46,7 @@ V přednášce MA3 jste již studovali tento typ rovnic, teď se však budeme věnovat obecnějšímu případu, kdy $A$ a $g$ závisí na $t$. -\begin{theorem} +\begin{theorem}[Globální existence a jednoznačnost] \label{thm-unique-sol-lineq} Nechť $t_0 \in (a, b), x_0 \in \mathbb{R}^n$ je dáno. Pak existuje jediné řešení rovnice \eqref{eq-linear-ode} definované na celém $(a, b)$ splňující počáteční podmínku $x(t_0) = x_0$. \end{theorem} @@ -62,7 +62,7 @@ V přednášce MA3 jste již studovali tento typ rovnic, teď se však budeme v Řešení $x$ splňuje $$ |x(t)| \leq |x(t_0)| + \int_{t_0}^t (\| A(s)\| |x(s)| + |g(s)|) ds \overset{\begin{subarray}{c}\|A(s)\| \leq L\\|g(s)| \leq \tilde{C}\\|x(t_0)|=C\end{subarray}}{\leq} C + \int_{t_0}^t (L|x(s)| + \tilde C) ds \leq $$ Z Gronwallova lemmatu dostaneme - $$ \leq C + \tilde{C}(\beta - t_0) + \int_{t_0}^t L|x(s)| ds \implies |x(t)| \leq \underbrace{C + \tilde C(\beta - t_0) e^{L(\beta - t_0)}}_{R}. $$ + $$ \leq C + \tilde{C}(\beta - t_0) + \int_{t_0}^t L|x(s)| ds \implies |x(t)| \leq \underbrace{(C + \tilde C(\beta - t_0)) e^{L(\beta - t_0)}}_{R}. $$ Došli jsme ke sporu s Větou \ref{thm-leaving-compact}, neboť řešení $x$ nemůže opustit kompakt $K$. \end{proof} @@ -78,7 +78,7 @@ Důležitá poznámka: řešení existuje globálně na oboru spojitosti $A(t), Použijeme znalosti lineární algebry k tomu, abychom mohli formalizovat postup řešení lineárních ODR. -\begin{theorem} +\begin{theorem}[Prostor řešení] Množina $\mathcal{R}_H$ řešení homogenní rovnice \eqref{eq-homogenous-linear-ode} bez zadané počáteční podmínky tvoří $n$-dimenzionální podprostor $C^1((a, b), \mathbb{R}^n)$. \end{theorem} diff --git a/skripta.pdf b/skripta.pdf index 9891097c5e91c4cb4fbbeadc17be075145a5445c..5957848fdc9d3fdc8433b88d71a06d97f1a75de8 100644 GIT binary patch delta 21985 zcmV)HK)t`*?hUx_4X~28e>pZjJ_>Vma%Ev{3V7P>TuYA}w-LVQS9ES(7)_JK;!9dD zuma)$x!KVz%3&WK0a6siK;TIKK2>~llRfO7p4F_Z7?H4;nI1N~s#uS&ip3dz8Pp)@ zzZx72P6ZM8-|+B(uDzwdKM${758wR|28BYSRlyFg4}(?0S+CtNe_9YhD!TXe>%RWOY_W;Ox=txYq3b9UkX>b^;=knSd>b1E6#~Q%zF>2%~Y*+hQ?bx)$SwC1b%j^j}xm)&GAo2pvdOaQM%lDCjt^$yts{ zIiZO{J$x7_+z3g7bba|B!(WFVbD(Hgt0w{SUD{y7B*;cR3zu4;R>oG^E-Bj45_r<* zX1yB$SV2JvkOXRtwK|Ucwkyu`R?Y5KU5(1x>77gOx|*j2G2}zO_uKB0w)3Ch7KC?R 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