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b/cviceni.tex @@ -34,4 +34,6 @@ \include{cviceni2} \include{cviceni3} +\include{zapocet1} + \end{document} diff --git a/zapocet1.tex b/zapocet1.tex new file mode 100644 index 0000000..949fb66 --- /dev/null +++ b/zapocet1.tex @@ -0,0 +1,34 @@ +\section{První zápočtová písemka} + +\begin{example} + Nalezněte řešení rovnice + $$ x^2 z \pdv{u}{x} + y^3 \pdv{u}{y} + z^2 \pdv{u}{z} = 0 $$ + splňující $u(x, y, 1) = e^x \cos y$ na okolí bodu $(1, 1, 1)$. +\end{example} + +\begin{proof} + Charakteristický systém je $\odv{x}{s} = x^2 z, \odv{y}{s} = y^3, \odv{z}{s} = z^2$. Z toho dostáváme $-\odv*{\frac{1}{x}}{s} = z = \odv*{\log |z|}{s}$, tedy $\odv*{(\log|z| + \frac{1}{x})}{s} = 0$, což je první charakteristika. Druhou nezávislou charakteristikou získáme zderivováním $\odv*{(\frac{1}{2y^2})}{s} = -1$ a $\odv*{(-\frac{1}{z})}{s} = 1$, tedy $\psi_2(x, y, z) = \frac{1}{z} - \frac{1}{2y^2}$. + + Budeme hledat řešení ve tvaru $u(x, y, z) = \psi(\log|z| + \frac{1}{x}, \frac{1}{z} - \frac{1}{2y^2})$. Dosadíme počáteční podmínku, dostaneme $u(x, y, 1) = e^x\cos y = \psi(\frac{1}{x}, 1 - \frac{1}{2y^2})$. Máme funkci $\psi(u, v) = e^\frac{1}{u}\cos\left(\sqrt{\frac{1}{2(1-v)}}\right)$. + + Zpětným dosazením získáme $u(x, y, z) = e^\frac{1}{\log z + \frac{1}{x}} \cos\left(\sqrt{\frac{1}{2(1 - \frac{1}{z} + \frac{1}{2y^2})}}\right)$. +\end{proof} + +\begin{example} + Nalezněte řešení rovnice + $$ \pdv{u}{x} + x\pdv{u}{y} = y + x$$ + splňující $u(1, y) = \cos y$ na jistém okolí bodu $(1, 1)$. +\end{example} + +\begin{proof} + Uvažujme homogenní rovnici + $$ \pdv{w}{x} + x\pdv{w}{y} + (x + y)\pdv{w}{z} = 0. $$ + Její charakteristický systém je $\odv{x}{s} = 1, \odv{y}{s} = x, \odv{z}{s} = x + y$. Z toho dostáváme $x = s + c_1$, $y = \frac{1}{2}s^2 + c_1s + c_2$, $z = \frac{1}{6}s^3 + \frac{c_1 + 1}{2}s^2 + (c_1 + c_2)s + c_3$. + + Fix $(x_0, y_0, z_0)$. Nalezneme hodnoty konstant tak, abychom se v čase $s = 0$ nacházeli v tomto bodě. Dostáváme $c_1 = x_0, c_2 = y_0, c_3 = z_0$. Dále hledáme $s$ takové, aby $x(s) = 1$. Takové $s = 1 - x_0$. Současně chceme, aby $u(1, y) = \cos y$. Platí $0 = z(s) - u(1, y(s)) = z(s) - \cos(y(s))$. Dostáváme + $$ 0 = z_0 + \frac{1}{6}(1 - x_0)^3 + \frac{x_0 + 1}{2}(1 - x_0)^2 + (x_0 + y_0)(1 - x_0) - $$ + $$ \cos\left(\frac{1}{2}(1 - x_0)^2 + x_0(1 - x_0) + y_0\right). $$ + Z toho již dostáváme + $$ u(x, y) = \cos\left(\frac{1}{2}x^2 + x(1 - x) + y\right) -\frac{1}{6}x^3 - \frac{1}{2}(x + 1)(1 - x)^2 - (x + y)(1 - x) $$ + na okolí bodu $(1, 1)$. +\end{proof}