% (find-LATEX "2024-1-C2-edolccs.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2024-1-C2-edolccs.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2024-1-C2-edolccs.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2024-1-C2-edolccs.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2024-1-C2-edolccs.pdf"))
% (defun e () (interactive) (find-LATEX "2024-1-C2-edolccs.tex"))
% (defun o () (interactive) (find-LATEX "2023-2-C2-edolccs.tex"))
% (defun u () (interactive) (find-latex-upload-links "2024-1-C2-edolccs"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2024-1-C2-edolccs.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
%          (code-eec-LATEX "2024-1-C2-edolccs")
% (find-pdf-page   "~/LATEX/2024-1-C2-edolccs.pdf")
% (find-sh0 "cp -v  ~/LATEX/2024-1-C2-edolccs.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2024-1-C2-edolccs.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2024-1-C2-edolccs.pdf")
%   file:///home/edrx/LATEX/2024-1-C2-edolccs.pdf
%               file:///tmp/2024-1-C2-edolccs.pdf
%           file:///tmp/pen/2024-1-C2-edolccs.pdf
%  http://anggtwu.net/LATEX/2024-1-C2-edolccs.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-cps   "Caepro5 Piecewise2 Maxima2 Escadas1")
% (find-Deps1-anggs "Caepro5 Piecewise2 Maxima2")
% (find-MM-aula-links "2024-1-C2-edolccs" "2" "c2m241edolccs" "c2el")

% «.defs»			(to "defs")
% «.defs-T-and-B»		(to "defs-T-and-B")
% «.defs-caepro»		(to "defs-caepro")
% «.defs-pict2e»		(to "defs-pict2e")
% «.defs-maxima»		(to "defs-maxima")
% «.defs-V»			(to "defs-V")
% «.title»			(to "title")
% «.links»			(to "links")
% «.raizes-chutar-testar»	(to "raizes-chutar-testar")



\documentclass[oneside,12pt]{article}
\usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage[x11names,svgnames]{xcolor} % (find-es "tex" "xcolor")
\usepackage{colorweb}                  % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (find-dn6 "preamble6.lua" "preamble0")
%\usepackage{proof}   % For derivation trees ("%:" lines)
%\input diagxy        % For 2D diagrams ("%D" lines)
%\xyoption{curve}     % For the ".curve=" feature in 2D diagrams
%
\usepackage{edrx21}               % (find-LATEX "edrx21.sty")
\input edrxaccents.tex            % (find-LATEX "edrxaccents.tex")
\input edrx21chars.tex            % (find-LATEX "edrx21chars.tex")
\input edrxheadfoot.tex           % (find-LATEX "edrxheadfoot.tex")
\input edrxgac2.tex               % (find-LATEX "edrxgac2.tex")
%
% (find-es "tex" "geometry")
\usepackage[a6paper, landscape,
            top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot
           ]{geometry}
%
\begin{document}

% «defs»  (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")

\def\drafturl{http://anggtwu.net/LATEX/2024-1-C2.pdf}
\def\drafturl{http://anggtwu.net/2024.1-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}

% (find-LATEX "2024-1-C2-carro.tex" "defs-caepro")
% (find-LATEX "2024-1-C2-carro.tex" "defs-pict2e")

\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"}  % (find-LATEX "dednat6load.lua")

% «defs-T-and-B»  (to ".defs-T-and-B")
\long\def\ColorDarkOrange#1{{\color{orange!90!black}#1}}
\def\T(Total: #1 pts){{\bf(Total: #1)}}
\def\T(Total: #1 pts){{\bf(Total: #1 pts)}}
\def\T(Total: #1 pts){\ColorRed{\bf(Total: #1 pts)}}
\def\B       (#1 pts){\ColorDarkOrange{\bf(#1 pts)}}

% «defs-caepro»  (to ".defs-caepro")
%L dofile "Caepro5.lua"              -- (find-angg "LUA/Caepro5.lua" "LaTeX")
\def\Caurl   #1{\expr{Caurl("#1")}}
\def\Cahref#1#2{\href{\Caurl{#1}}{#2}}
\def\Ca      #1{\Cahref{#1}{#1}}

% «defs-pict2e»  (to ".defs-pict2e")
%L dofile "Piecewise2.lua"           -- (find-LATEX "Piecewise2.lua")
%L dofile "Escadas1.lua"           -- (find-LATEX "Escadas1.lua")
\def\pictgridstyle{\color{GrayPale}\linethickness{0.3pt}}
\def\pictaxesstyle{\linethickness{0.5pt}}
\def\pictnaxesstyle{\color{GrayPale}\linethickness{0.5pt}}
\celllower=2.5pt

% «defs-maxima»  (to ".defs-maxima")
%L dofile "Maxima2.lua"              -- (find-angg "LUA/Maxima2.lua")
\pu

% «defs-V»  (to ".defs-V")
%L --- See: (find-angg "LUA/MiniV1.lua" "problem-with-V")
%L V = MiniV
%L v = V.fromab
\pu


%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (c2m241edolccsp 1 "title")
% (c2m241edolccsa   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 2 - 2024.1}

\bsk

Aula 30: EDOLCCs

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2024.1-C2.html}

\end{center}

\newpage

% «links»  (to ".links")
% (c2m241edolccsp 2 "links")
% (c2m241edolccsa   "links")

{\bf Links}

\scalebox{0.5}{\def\colwidth{16cm}\firstcol{

% (find-books "__analysis/__analysis.el" "stewart-pt" "1020" "17.1 Equações Lineares de Segunda Ordem")
% (find-books "__analysis/__analysis.el" "stewart-pt" "1034" "subamortecimento")
% (find-books "__analysis/__analysis.el" "stewart-pt" "51" "H Números Complexos")
\par \Ca{StewPtCap17p6} (p.1020) Equações diferenciais de 2ª ordem
\par \Ca{StewPtCap17p20} (p.1034) Caso 3: subamortecimento
\par \Ca{StewPtApendiceHp5} (p.A51) Apêndice H: Números complexos

\ssk

% (find-books "__analysis/__analysis.el" "leithold" "156" "3.3. Teoremas sobre derivação")
\par \Ca{Leit3p22} (p.158) $D_x[c·f(x)] = c·D_xf(x)$
\par \Ca{Leit3p22} (p.158) $D_x[f(x)+g(x)] = D_xf(x)+D_xg(x)$

\ssk

% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "105" "3. Equações lineares de segunda")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "111" "operador diferencial")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "113" "princípio da superposição")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "121" "3.3. Raízes complexas")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "123" "Figura 3.3.1")
\par \Ca{BoyceDip3p5} (p.105) Capítulo 3: Equações lineares de 2ª ordem
\par \Ca{BoyceDip3p11} (p.111) Seção 3.2: o operador diferencial $L$
\par \Ca{BoyceDip3p13} (p.113) Teorema 3.2.2: o princípio da superposição
\par \Ca{BoyceDip3p21} (p.121) 3.3. Raízes complexas da equação característica
\par \Ca{BoyceDip3p23} (p.123) Figura 3.3.1

\ssk

% (find-books "__analysis/__analysis.el" "zill-cullen-pt" "173" "4.3" "coeficientes constantes")
% (find-books "__analysis/__analysis.el" "zill-cullen-pt" "196" "Exemplo 1: ...pode ser fatorado...")
% (find-books "__analysis/__analysis.el" "zill-cullen" "150" "FACTORING OPERATORS")
\par \Ca{ZillCullenCap4p33} (p.173) 4.3. Equações lineares homogêneas com coeficientes constantes
\par \Ca{ZillCullenCap4p60} (p.196) Exemplo 1: ...pode ser fatorado... $(D+3)(D+2)$
\par \Ca{ZillCullenCap4p61} (p.197) Exemplo 3
\par \Ca{ZillCullenCap4p64} (p.200) Exercícios
\par \Ca{ZillCullenEngCap4p40} (p.150) Factoring operators ... $(D+3)(D+2)$

\msk

% (find-books "__analysis/__analysis.el" "boyce-diprima" "103" "3 Second-Order Linear")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "110" "differential operator")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "112" "Theorem 3.2.2" "Superposition")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "120" "3.3 Complex Roots")
\par \Ca{BoyceDipEng3p4} (p.103) Chapter 3: Second-order linear ODEs
\par \Ca{BoyceDipEng3p11} (p.110) Section 3.2: the differential operator $L$
\par \Ca{BoyceDipEng3p13} (p.112) Theorem 3.2.2: principle of superposition
\par \Ca{BoyceDipEng3p21} (p.120) 3.3 Complex Roots of the Characteristic Equation
\par \Ca{BoyceDipEng3p24} (p.123) Figure 3.3.1

\ssk

Quadros:

% (find-angg ".emacs" "c2q232" "30,nov01: EDOLCCs")
\par \Ca{2hQ61} Aula 30 de 2023.2 (01/nov/2023)

\standout{Aviso:} falta digitar muita coisa! Veja os quadros!

% (c2m231dicasp2p 4 "raizes-chutar-testar")
% (c2m231dicasp2a   "raizes-chutar-testar")
% 2gT128 Raízes por chutar-e-testar

\bsk
\bsk

Quadros antigos:
% (find-angg ".emacs" "c2q231" "jun20: EDOs lineares")
\par \Ca{2gQ46} Aula 23 de 2023.1 (20/junho/2023)
\par \Ca{2gQ50} Aula 24 de 2023.1 (23/junho/2023)

\bsk

}\anothercol{
}}

\newpage


% (c2m231edolsp 3 "nao-usei")
% (c2m231edolsa   "nao-usei")

\def\Mscale{0.5}
\def\dqeq{\;\text{``$=$''}\;}
\def\M#1{\scalebox{\Mscale}{$\pmat{#1}$}}

$\M{0&1\\0&0} \M{0&0\\1&0} = \M{1&0\\0&0}$

\ssk

$\M{0&0\\1&0} \M{0&1\\0&0} = \M{0&0\\0&1}$

\msk

$\M{a&b} \M{c\\d} = \M{ac+bd}$

$\M{c\\d} \M{a&b} = \M{ac&bc\\ad&bd}$

\msk

$S = \M{0&1 \\ &0&1 \\ &&0&1 \\ &&&0&1 \\ &&&&0}
 \quad
 1 = \M{1 \\ &1 \\ &&1 \\ &&&1 \\ &&&&1}
 \quad
 S-1 = \M{-1&1 \\ &-1&1 \\ &&-1&1 \\ &&&-1&1 \\ &&&&-1}$

\msk

$v = \M{v_1\\v_2\\v_3\\v_4\\v_5\\}
 \quad
 f = \M{f(1)\\f(2)\\f(3)\\f(4)\\f(5)\\}
 \quad
 Sf = \M{f(2)\\f(3)\\f(4)\\f(5)\\0}
 \quad
 (S-1)f = \M{f(2)-f(1)\\f(3)-f(2)\\f(4)-f(3)\\f(5)-f(4)\\0-f(5)\\}
$

\bsk

Obs: não usei isso aqui --

não deu tempo de \LaTeX ar tudo...

\newpage

% (c2m232p1p 4 "questao-5-grids")
% (c2m232p1a   "questao-5-grids")

%L fry = FromYs.from {ys={0,-1,1,-2,2,-3,3,-3,2,-2,1,-1,0},  Y0=0} :setall()
%L fry = FromYs.from {ys={0,-1,-3,3,1,0,1,2,1,0,-1,-2,-1,0}, Y0=0} :setall()
%L fry = FromYs.from {ys={2,1,0,1,2,-2,1,-2,0,-2,0,1,2,1,0,-1,-2,-1,0}, Y0=-3} :setall()
%L Pict {
%L   fry:ypict()   :prethickness("1pt"):sa("fig f"),
%L   fry:Ypict()   :prethickness("1pt"):sa("fig F"),
%L   fry:grid(-4,4):sa("grid F"),
%L } :output()
\pu

\unitlength=10pt

$\begin{array}{rcl}
  g(x) &=& \ga{fig F} \\ \\[-5pt]
 g'(x) &=& \ga{fig f} \\
 \end{array}
$

\newpage

%M (%i1) f   : exp( 3*x);
%M (%o1) e^{3\,x}
%M (%i2) f   : exp(-3*x);
%M (%o2) e^ {- 3\,x }
%M (%i3) f   : exp( 2*x);
%M (%o3) e^{2\,x}
%M (%i4) fp  : diff(f,x);
%M (%o4) 2\,e^{2\,x}
%M (%i5) fpp : diff(f,x,2);
%M (%o5) 4\,e^{2\,x}
%M (%i6) Lf  : fpp + fp - 6*f;
%M (%o6) 0
%M (%i7) 
%L maximahead:sa("L1", "")
\pu

%M (%i1) D (f) := diff(f,x);
%M (%o1) D\left(f\right):=\mathrm{diff}\left(f , x\right)
%M (%i2) DD(f) := diff(f,x,2);
%M (%o2) \mathrm{DD}\left(f\right):=\mathrm{diff}\left(f , x , 2\right)
%M (%i3) L (f) := D(D(f)) + D(f) - 6*f;
%M (%o3) L\left(f\right):=D\left(D\left(f\right)\right)+D\left(f\right)+\left(-6\right)\,f
%M (%i4)   D(x^2);
%M (%o4) 2\,x
%M (%i5) D(D(x^2));
%M (%o5) 2
%M (%i6) L(x^2);
%M (%o6) -\left(6\,x^2\right)+2\,x+2
%M (%i7) L(exp( 3*x));
%M (%o7) 6\,e^{3\,x}
%M (%i8) L(exp(-3*x));
%M (%o8) 0
%M (%i9) L(exp( 2*x));
%M (%o9) 0
%M (%i10) 
%L maximahead:sa("L2", "")
\pu


\scalebox{0.4}{\def\colwidth{9cm}\firstcol{

\ga{L1}

}\anothercol{

\def\hboxthreewidth {10cm}
\ga{L2}

}}

\newpage

% «solucoes-nao-basicas»  (to ".solucoes-nao-basicas")
% (c2m232edolccsp 6 "solucoes-nao-basicas")
% (c2m232edolccsa   "solucoes-nao-basicas")

{\bf Soluções não-básicas}

% (find-c2q232page 64 "30,nov01: EDOLCCs")

\def\und      #1#2{\underbrace{#1}_{#2}}
\def\uuund#1#2#3#4{\und{\und{\und{#1}{#2}}{#3}}{#4}}


\scalebox{0.8}{\def\colwidth{12cm}\firstcol{

$$\begin{array}{rcl}
  M(αv + βw) &=& M(αv) + M(βw) \\
             &=& α(Mv) + β(Mw) \\
  \end{array}
$$

$$\und{(D-2)(D+3)}{M}
  (\und{42}{α} \und{e^{2x}}{v} + \und{99}{β} \und{e^{-3x}}{w})
$$

\sa{(D-2)e^{2x}}{
  \uuund{(D-2)e^{2x}}
       {De^{2x} -2e^{2x}}
       {2e^{2x} -2e^{2x}}
       {0}
  }
\sa{(D+3)e^{-3x}}{
  \uuund{(D+3)e^{-3x}}
       {De^{-3x}+3e^{-3x}}
       {-3e^{-3x}+3e^{-3x}}
       {0}
  }

$$\begin{array}{l}
  (D-2)(D+3)(42e^{2x} + 99e^{-3x}) \\
  = \; 42(D-2)(D+3)e^{2x} + 99(D-2)(D+3)e^{-3x} \\
  = \; \und{42\und{(D+3)\ga{(D-2)e^{2x}}}{0}}{0}
    \,+\, \und{99\und{(D-2)\ga{(D+3)e^{-3x}}}{0}}{0} \\
  \end{array}
$$

}\anothercol{
}}


\newpage

% «raizes-chutar-testar»  (to ".raizes-chutar-testar")
% (c2m241edolccsp 8 "raizes-chutar-testar")
% (c2m241edolccsa   "raizes-chutar-testar")
% (find-es "maxima" "divisors")

%M (%i1) poly0 : (x-2)*(x+5);
%M (%o1) \left(x-2\right)\,\left(x+5\right)
%M (%i2) poly1 : expand(poly0);
%M (%o2) x^2+3\,x-10
%M (%i3) b : ratcoef(poly1, x, 1);    /* coeficiente do x^1 */
%M (%o3) 3
%M (%i4) c : ratcoef(poly1, x, 0);    /* coeficiente do x^0 */
%M (%o4) -10
%M (%i5)                 divisors(c);
%M (%o5) \left \{1 , 2 , 5 , 10 \right \}
%M (%i6) divs0 : listify(divisors(c));
%M (%o6) \left[ 1 , 2 , 5 , 10 \right] 
%M (%i7)                 reverse(divs0);
%M (%o7) \left[ 10 , 5 , 2 , 1 \right] 
%M (%i8)                -reverse(divs0);
%M (%o8) \left[ -10 , -5 , -2 , -1 \right] 
%M (%i9) divs1 : append(-reverse(divs0), divs0);
%M (%o9) \left[ -10 , -5 , -2 , -1 , 1 , 2 , 5 , 10 \right] 
%L maximahead:sa("raizes", "")
\pu

%M (%i10) line(d1) := block([d2:c/d1], [d1,d2,d1*d2,d1+d2])$
%M (%i11) line(2);
%M (%o11) \left[ 2 , -5 , -10 , -3 \right] 
%M (%i12) lines0 : makelist(line(d1), d1, divs1)$
%M (%i13) lines1 : append([["d1", "d2", "d1*d2", "d1+d2"]], lines0)$
%M (%i14) apply('matrix, lines1);
%M (%o14) \begin{pmatrix}\mbox{ d1 }&\mbox{ d2 }&\mbox{ d1*d2 }&\mbox{ d1+d2 }\cr -10&1&-10&-9\cr -5&2&-10&-3\cr -2&5&-10&3\cr -1&10&-10&9\cr 1&-10&-10&-9\cr 2&-5&-10&-3\cr 5&-2&-10&3\cr 10&-1&-10&9\cr \end{pmatrix}
%M (%i15) 
%L maximahead:sa("raizes 2", "")
\pu

{\bf Raízes por chutar-e-testar}

\scalebox{0.375}{\def\colwidth{14cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{raizes}

}\anothercol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{raizes 2}

}}




\GenericWarning{Success:}{Success!!!}  % Used by `M-x cv'

\end{document}



% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2el"
% ee-tla: "c2m241edolccs"
% End: