%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Wednesday, February 10, 1999 13:29:48} %TCIDATA{LastRevised=Sunday, February 13, 2005 19:04:10} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za I online - Funkcie - Cvi\U{10d}enia\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \author{A. U. Thor} \title{Lab Report} \date{The Date } \maketitle \begin{abstract} A Laboratory report created with Scientific Notebook \end{abstract} \section{Laplaceova transform\'{a}cia} \begin{center} \begin{tabular}{|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{mcindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{mcindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{K4.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{K4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{Ot4.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{Ot4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Ind.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Ind.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \section{Cvi\v{c}enia} \begin{enumerate} \item Zistite, \v{c}i $f\left( t\right) $\ je origin\'{a}l a n\'{a}jdite jeho Laplaceovu transform\'{a}ciu \begin{enumerate} \item $f\left( t\right) =\left\{ \begin{tabular}{cc} $0$ & $t<0$ \\ $\frac{3}{t-5}$ & $t\geq 0$% \end{tabular}% \right. .% \CustomNote{Answer}{$f\left( t\right) $ nesp\'{l}\v{n}a podmienku (3), teda nie je origin\'{a}l.}$ \item $f\left( t\right) =\left\{ \begin{tabular}{cc} $0$ & $t<0$ \\ $e^{5t}$ & $t\geq 0$% \end{tabular}% \right. .% \CustomNote{Answer}{$\frac{1}{p-5}.$}$ \item $f\left( t\right) =\left\{ \begin{tabular}{cc} $0$ & $t<0$ \\ $e^{t^{2}}$ & $t\geq 0$% \end{tabular}% \right. .% \CustomNote{Answer}{$f\left( t\right) $ nesp\'{l}\v{n}a podmienku (3), teda nie je origin\'{a}l.}$ \item $f\left( t\right) =e^{3t},$ \CustomNote{Answer}{$\frac{1}{p-3}.$} \item $f\left( t\right) =\sin 2t,$ \CustomNote{Answer}{$\frac{2}{p^{2}+4}.$} \item $f\left( t\right) =\cos 5t,$ \CustomNote{Answer}{$\frac{p}{p^{2}+25}.$} \item $f\left( t\right) =t^{5},$ \CustomNote{Answer}{$\frac{5!}{p^{6}}.$} \item $f\left( t\right) =\sin ^{2}\omega t,$ \CustomNote{Answer}{$\frac{2\omega ^{2}}{p\left( p^{2}+4\omega ^{2}\right) }% . $} \item $f\left( t\right) =\cos ^{2}\omega t,$ \CustomNote{Answer}{$\frac{p^{2}+2\omega ^{2}}{p\left( p^{2}+4\omega ^{2}\right) }.$} \item $f\left( t\right) =e^{4t}\cos t,$ \CustomNote{Answer}{$\frac{p-4}{p^{2}-8p+17}.$} \item $f\left( t\right) =e^{2t}\sin 3t,$ $% \CustomNote{Answer}{$\frac{3}{p^{2}-4p+13}.$}$ \item $f\left( t\right) =3t\cos 4t,$ \CustomNote{Answer}{$\frac{3p^{2}-48}{\left( p^{2}+16\right) ^{2}}.$} \item $f\left( t\right) =\sin 2t-2t\cos 2t,$ $% \CustomNote{Answer}{$\frac{16}{\left( p^{2}+4\right) ^{2}}.$}$ \item $f\left( t\right) =\eta \left( t-3\right) \sin 2\left( t-3\right) ,$ \CustomNote{Answer}{$e^{-3p}\frac{2}{p^{2}+4}.$} \item $f\left( t\right) =\eta \left( t-1\right) e^{2\left( t-1\right) }\cos 3\left( t-1\right) .$ \CustomNote{Answer}{$e^{-p}\frac{p-2}{\left( p-2\right) ^{2}+9}.$} \end{enumerate} \item N\'{a}jdite origin\'{a}ly k Laplaceov\'{y}m obrazom \begin{enumerate} \item $F\left( p\right) =\frac{1}{p^{2}},\,% \CustomNote{Answer}{$\frac{t^{4}}{4!}.$}$ \item $F\left( p\right) =\frac{1}{\left( p-3\right) ^{4}},\,% \CustomNote{Answer}{$\frac{t^{3}}{3!}e^{3t}.$}$ \item $F\left( p\right) =\frac{3}{p^{2}+9},\,% \CustomNote{Answer}{$\sin 3t.$}$ \item $F\left( p\right) =\frac{3p}{p^{2}-16},\,% \CustomNote{Answer}{$3\cosh 4t.$}$ \item $F\left( p\right) =\frac{4}{\left( p-3\right) ^{2}-4},\,% \CustomNote{Answer}{$2e^{3t}\sinh 2t.$}$ \item $F\left( p\right) =\frac{p-1}{p^{2}-2p+10},\,% \CustomNote{Answer}{$e^{t}\cos 3t.$}$ \item $F\left( p\right) =\frac{3e^{-p}}{p^{2}+9},\,% \CustomNote{Answer}{$\eta \left( t-1\right) \sin 3\left( t-1\right) .$}$ \item $F\left( p\right) =\frac{1}{p^{10}},\,% \CustomNote{Answer}{$\frac{t^{9}}{9!}.$}$ \item $F\left( p\right) =\frac{1}{\left( p-3\right) ^{4}},\,% \CustomNote{Answer}{$e^{3t}\frac{t^{3}}{3!}.$}$ \item $F\left( p\right) =e^{-2p}\frac{3}{p^{2}-9},\,% \CustomNote{Answer}{$\eta \left( t-2\right) \sinh 3\left( t-2\right) .$}$ \item $F\left( p\right) =\frac{p+1}{p^{2}+2p},\,% \CustomNote{Answer}{$e^{-t}\cosh t.$}$ \item $F\left( p\right) =\frac{\omega }{p\left( p^{2}+\omega ^{2}\right) },\,% \CustomNote{Answer}{$\frac{1}{\omega }\left( 1-\cos \omega t\right) .$}$ \item $F\left( p\right) =\frac{1}{\left( p^{2}+\omega ^{2}\right) ^{2}},\,% \CustomNote{Answer}{$\frac{1}{2\omega ^{2}}\left( \sin \omega t-\omega t\cos \omega t\right) .$}$ \item $F\left( p\right) =\frac{4}{p^{2}\left( p^{2}+16\right) },\,% \CustomNote{Answer}{$\frac{t}{4}-\frac{\sin 4t}{16}.$}$ \item $F\left( p\right) =\frac{3p}{\left( p-1\right) \left( p-3\right) \left( p-4\right) },\,% \CustomNote{Answer}{$\frac{1}{2}e^{t}-\frac{9}{2}e^{3t}+4e^{4t}.$}$ \item $F\left( p\right) =\frac{6p-14}{p^{2}-4p+3},\,% \CustomNote{Answer}{$2e^{3t}+4e^{t}.$}$ \item $F\left( p\right) =\frac{2p^{3}+9p^{2}+15p+8}{p\left( p+2\right) \left( p+1\right) ^{2}},\,% \CustomNote{Answer}{$4+e^{-2t}-3e^{-t}.$}$ \item $F\left( p\right) =\frac{p^{2}}{\left( p^{2}+1\right) ^{2}}.\,% \CustomNote{Answer}{$\cos t-\frac{1}{2}t\sin t.$}$ \end{enumerate} \item Rie\v{s}te nasleduj\'{u}ce Z\'{U}: \begin{enumerate} \item $y\,^{\prime \prime }-3y\,^{\prime }+2y=te^{t},\,y\left( 0+\right) =1,\,y\,^{\prime }\left( 0+\right) =-2.\,% \CustomNote{Answer}{$y\left( t\right) =-\frac{t^{2}}{2}% e^{t}-te^{t}+3e^{t}-2e^{2t}.$}$ \item $y\,^{\prime \prime \prime }+4y\,^{\prime }=1,\,y\left( 0+\right) =y\,^{\prime }\left( 0+\right) =y\,^{\prime \prime }\left( 0+\right) =0.\,% \CustomNote{Answer}{$y\left( t\right) =-\frac{t}{4}-\frac{1}{8}\sin 2t.$}$ \item $y\,^{\prime \prime }-4y\,^{\prime }+5y=0,\,y\left( 0+\right) =0,\,y\,^{\prime }\left( 0+\right) =1.\,% \CustomNote{Answer}{$y\left( t\right) =e^{2t}\sin t.$}$ \item $y\,^{\prime \prime }-4y\,^{\prime }+3y=0,\,y\left( 0+\right) =6,\,y\,^{\prime }\left( 0+\right) =10.\,% \CustomNote{Answer}{$y\left( t\right) =2e^{3t}+4e^{t}.$}$ \item $y\,^{\prime \prime }+4y\,^{\prime }+29y=0,\,y\left( 0+\right) =0,\,y\,^{\prime }\left( 0+\right) =15.\,% \CustomNote{Answer}{$y\left( t\right) =3e^{-2t}\sin 5t.$}$ \item $y\,^{\prime \prime }+2y\,^{\prime }+2y=2+2t,\,y\left( 0+\right) =0,\,y\,^{\prime }\left( 0+\right) =1.% \CustomNote{Answer}{$y\left( t\right) =t.$}\;$ \item $y\,^{\prime \prime }-y=t,\,y\left( 0+\right) =0,\,y\,^{\prime }\left( 0+\right) =1.\;% \CustomNote{Answer}{$y\left( t\right) =2\sinh t-t.$}$ \item $y\,^{\prime \prime \prime }+2y\,^{\prime \prime }+y\,^{\prime }=-2e^{-2t},\,y\left( 0+\right) =2,\,y\,^{\prime }\left( 0+\right) =1,\,y\,^{\prime \prime }\left( 0+\right) =1.\;% \CustomNote{Answer}{$y\left( t\right) =4+e^{-2t}-3e^{-t}.$}$ \end{enumerate} \item Rie\v{s}te nasleduj\'{u}ce Z\'{U} pre syst\'{e}my rovn\'{\i}c: \begin{enumerate} \item $y_{1}^{\prime }=y_{2},\,y_{2}^{\prime }=y_{1}+t,\,y_{1}\left( 0+\right) =0,\,y_{2}\left( 0+\right) =0.\,% \CustomNote{Answer}{$y_{1}\left( t\right) =\sinh t-t,$% \par $y_{2}\left( t\right) =\cosh t-1.$}$ \item $y_{1}^{\prime }=y_{2}+y_{3},\,y_{2}^{\prime }=y_{1}+y_{3},\,y_{3}^{\prime }=y_{1}+y_{2},\,\,y_{1}\left( 0+\right) =-1,\,y_{2}\left( 0+\right) =1,\,y_{3}\left( 0+\right) =0.\,% \CustomNote{Answer}{$y_{1}\left( t\right) =-e^{-t},$% \par $y_{2}\left( t\right) =e^{-t},\,$% \par $y_{3}\left( t\right) =0.$}$ \item $y_{1}^{\prime }=-3y_{1}+4y_{2}+9e^{2t},\,y_{2}^{\prime }=-2y_{1}+3y_{2}+e^{2t},\,y_{1}\left( 0+\right) =2,\,y_{2}\left( 0+\right) =0.\,% \CustomNote{Answer}{$y_{1}\left( t\right) =e^{t}+e^{2t},$% \par $y_{2}\left( t\right) =e^{t}-e^{2t}.$}$ \item $y_{1}^{\prime }=2y_{1}+4y_{2}+\cos t,\,y_{2}^{\prime }=-y_{1}-2y_{2}+\sin t,\,y_{1}\left( 0+\right) =0,\,y_{2}\left( 0+\right) =0.\,% \CustomNote{Answer}{$y_{1}\left( t\right) =4t+2-2\cos t-3\sin t,$% \par $y_{2}\left( t\right) =-2t+2\sin t.$}$ \end{enumerate} \end{enumerate} \begin{center} \begin{tabular}{|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{mcindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{mcindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{K4.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{K4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{Ot4.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{Ot4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Ind.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Ind.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za III} \section{Laplaceova transform\'{a}cia} \end{document}