% (find-LATEX "2023-2-C2-edos-exatas.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2023-2-C2-edos-exatas.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2023-2-C2-edos-exatas.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2023-2-C2-edos-exatas.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2023-2-C2-edos-exatas.pdf"))
% (defun e () (interactive) (find-LATEX "2023-2-C2-edos-exatas.tex"))
% (defun o () (interactive) (find-LATEX "2023-2-C2-edos-exatas.tex"))
% (defun u () (interactive) (find-latex-upload-links "2023-2-C2-edos-exatas"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2023-2-C2-edos-exatas.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
%          (code-eec-LATEX "2023-2-C2-edos-exatas")
% (find-pdf-page   "~/LATEX/2023-2-C2-edos-exatas.pdf")
% (find-sh0 "cp -v  ~/LATEX/2023-2-C2-edos-exatas.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2023-2-C2-edos-exatas.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2023-2-C2-edos-exatas.pdf")
%   file:///home/edrx/LATEX/2023-2-C2-edos-exatas.pdf
%               file:///tmp/2023-2-C2-edos-exatas.pdf
%           file:///tmp/pen/2023-2-C2-edos-exatas.pdf
%  http://anggtwu.net/LATEX/2023-2-C2-edos-exatas.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-cps   "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-anggs "Caepro5 Piecewise2 Maxima2")
% (find-MM-aula-links "2023-2-C2-edos-exatas" "C2" "c2m232edosexatas" "c2ee")

% «.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-edoexs»		(to "defs-edoexs")
% «.title»			(to "title")
% «.links»			(to "links")
% «.metodo-e-exemplo»		(to "metodo-e-exemplo")
% «.uma-questao-de-prova»	(to "uma-questao-de-prova")
% «.maxima»			(to "maxima")
% «.defs-e-exemplos»		(to "defs-e-exemplos")
%
% «.djvuize»			(to "djvuize")



\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")
%\usepackage{emaxima}              % (find-LATEX "emaxima.sty")
%
% (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/2023-2-C2.pdf}
\def\drafturl{http://anggtwu.net/2023.2-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}

% (find-LATEX "2023-1-C2-carro.tex" "defs-caepro")
% (find-LATEX "2023-1-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-edoexs»  (to ".defs-edoexs")
\input 2023-2-C2-edos-exatas-defs.tex % (find-LATEX "2023-2-C2-edos-exatas-defs.tex")




%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (c2m232edosexatasp 1 "title")
% (c2m232edosexatasa   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo C2 - 2023.2}

\bsk

Aula 34: EDOs exatas

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2023.2-C2.html}

\end{center}

\newpage

% «links»  (to ".links")
% (c2m232edosexatasp 2 "links")
% (c2m232edosexatasa   "links")

{\bf Links}

\scalebox{0.6}{\def\colwidth{15cm}\firstcol{

% (find-books "__analysis/__analysis.el" "stewart-pt" "811" "14.3 Derivadas Parciais")
% (find-books "__analysis/__analysis.el" "stewart-pt" "831" "14.5 A Regra da Cadeia")
% (find-books "__analysis/__analysis.el" "stewart-pt" "833" "A Regra da Cadeia (versão geral)")
\par \Ca{StewPtCap14p25} (p.811) 14.3 Derivadas Parciais
\par \Ca{StewPtCap14p45} (p.831) 14.5 A Regra da Cadeia
\par \Ca{StewPtCap14p47} (p.833) A Regra da Cadeia (versão geral)

\ssk

% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "72" "2.6. Equações Exatas")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "76" "Problemas")
% (find-books "__analysis/__analysis.el" "boyce-diprima"    "70" "2.6 Exact Differential Equations")
% (find-books "__analysis/__analysis.el" "boyce-diprima"    "75" "Problems")
\par \Ca{BoyceDip2p54} (p.72) 2.6 Equações Exatas e Fatores Integrantes
\par \Ca{BoyceDip2p58} (p.76) Problemas
\par \Ca{BoyceDipEng2p50} (p.70) 2.6 Exact Differential Equations and Integrating Factors
\par \Ca{BoyceDipEng2p55} (p.75) Problems

\ssk

% (find-books "__analysis/__analysis.el" "zill-cullen-pt" "60" "2.4. Equações exatas")
% (find-books "__analysis/__analysis.el" "zill-cullen-pt" "67" "Exercícios")
% (find-books "__analysis/__analysis.el" "zill-cullen"    "62" "2.4. Exact Equations")
% (find-books "__analysis/__analysis.el" "zill-cullen"    "68" "Exercises 2.4")
\par \Ca{ZillCullenCap2p25} (p.60) 2.4 Equações exatas
\par \Ca{ZillCullenCap2p32} (p.67) Exercícios
\par \Ca{ZillCullenEngCap2p35} (p.62) 2.4 Exact equations
\par \Ca{ZillCullenEngCap2p41} (p.68) Exercises 2.4

\ssk

% (find-books "__analysis/__analysis.el" "lebl" "63" "1.8 Exact equations")
% (find-books "__analysis/__analysis.el" "lebl" "70" "1.8.3 Exercises")
\par \Ca{DiffyQsP63} 1.8 Exact Equations
\par \Ca{DiffyQsP70} 1.8.3 Exercises

\msk

% (c2m222edoexsp 2 "links")
% (c2m222edoexsa   "links")
% (c2m192p2p)
% (c2m192p2a)
\par \Ca{2yT14} (2019.2) P2
\par \url{http://angg.twu.net/LATEX/2019-2-C2-P2.pdf}

% (find-angg ".emacs" "c2q232" "EDOs exatas")
% (find-angg ".emacs" "c2q222" "EDOs exatas por chutar e testar")
% (find-angg ".emacs" "c2q192" "22: EDOs exatas")
% (find-angg ".emacs" "c2q191" "EDOs exatas")
\par \Ca{2hQ73} (2023.2) Quadros das aulas sobre EDOs exatas
\par \Ca{2yQ106} (2019.2) Quadros das aulas sobre EDOs exatas

}\anothercol{
}}

\newpage

% «metodo-e-exemplo»  (to ".metodo-e-exemplo")
% (c2m232edosexatasp 3 "metodo-e-exemplo")
% (c2m232edosexatasa   "metodo-e-exemplo")
% (c2m232edovsp 6 "defs-e-exemplos")
% (c2m232edovsa   "defs-e-exemplos")

{\bf Método e exemplo}

\vspace*{0.5cm}

\scalebox{0.6}{\def\colwidth{8cm}\firstcol{

$\ga{reset}
  \begin{array}[t]{rcl}
  \ga{[E5]} &=& \ga{(E5)}  \\ \\[-5pt]
  \ga{[E3]} &=& \ga{(E3)} \\ \\[-5pt]
  \ga{[E2]} &=& \ga{(E2)} \\ \\[-5pt]
  \ga{[S1]} &=& \ga{reset-S1}
                \ga{(S)}  \\
  \end{array}
$

}\anothercol{

$\ga{reset-S1}
  \begin{array}[t]{rcl}
    \ga{[E5]}\ga{[S1]} &=& \ga{(E5)}  \\ \\[-5pt]
    \ga{[E3]}\ga{[S1]} &=& \ga{(E3)} \\ \\[-5pt]
    \ga{[E2]}\ga{[S1]} &=& \ga{(E2)} \\
  \end{array}
$
}}


\newpage

% «uma-questao-de-prova»  (to ".uma-questao-de-prova")
% (c2m232edosexatasp 5 "uma-questao-de-prova")
% (c2m232edosexatasa   "uma-questao-de-prova")

{\bf Uma questão da P2 de 2019.2}

\sa{3*}{\ensuremath{({*}{*}{*})}}
\sa{4*}{\ensuremath{({*}{*}{*}{*})}}

\scalebox{0.7}{\def\colwidth{10cm}\firstcol{

4) Sejam $\ga{3*}$ e $\ga{4*}$ estas EDOs:
%
$$\begin{array}{rl}
    2xy^3\,dx + 3x^2y^2\,dy = 0 & \ga{3*} \\
  2x^2y^3\,dx + 3x^3y^2\,dy = 0 & \ga{4*} \\
  \end{array}
$$

a) \B(0.5 pts) Mostre que $\ga{3*}$ é exata.

b) \B(0.5 pts) Encontre a solução geral de $\ga{3*}$.

c) \B(1.0 pts) Teste a sua solução geral da $\ga{3*}$.

d) \B(0.5 pts) Mostre que a solução geral da EDO $\ga{3*}$ também é
solução da $\ga{4*}$.

e) \B(0.5 pts) Mostre que $\ga{4*}$ não é exata.

f) \B(0.5 pts) Mostre que o fator integrante obtido por
%
$$\begin{array}{rcl}
     p(x) &=& (M_y - N_x) / N, \\
   \mu(x) &=& e^{\intx{p(x)}} \\
  \end{array}
$$
%
transforma $\ga{4*}$ em $\ga{3*}$.

\bsk

Versão original:
% (c2m192p2p)
% (c2m192p2a)
% \par \Ca{2yT14} (2019.2) P2
\par \url{http://angg.twu.net/LATEX/2019-2-C2-P2.pdf}

\bsk


}\anothercol{
}}

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

\end{document}


% «maxima»  (to ".maxima")

 (eepitch-maxima)
 (eepitch-kill)
 (eepitch-maxima)
[Mx,My]   : [2*x  *y^3, 3*x^2*y^2];
[Mxy,Myx] : [diff(Mx,y), diff(My,x)];

[Mx,My]   : [2*x^2*y^3, 3*x^3*y^2];
[Mxy,Myx] : [diff(Mx,y), diff(My,x)];

define(p(x), (Mxy-Myx)/My);
                  integrate(p(x),x);
              exp(integrate(p(x),x));
define(mu(x), exp(integrate(p(x),x)));
mu(x) * [Mx,My];


%  ____  _             _         
% |  _ \(_)_   ___   _(_)_______ 
% | | | | \ \ / / | | | |_  / _ \
% | |_| | |\ V /| |_| | |/ /  __/
% |____// | \_/  \__,_|_/___\___|
%     |__/                       
%
% «djvuize»  (to ".djvuize")
% (find-LATEXgrep "grep --color -nH --null -e djvuize 2020-1*.tex")

 (eepitch-shell)
 (eepitch-kill)
 (eepitch-shell)
# (find-fline "~/2023.2-C2/")
# (find-fline "~/LATEX/2023-2-C2/")
# (find-fline "~/bin/djvuize")

cd /tmp/
for i in *.jpg; do echo f $(basename $i .jpg); done

f () { rm -v $1.pdf;  textcleaner -f 50 -o  5 $1.jpg $1.png; djvuize $1.pdf; xpdf $1.pdf }
f () { rm -v $1.pdf;  textcleaner -f 50 -o 10 $1.jpg $1.png; djvuize $1.pdf; xpdf $1.pdf }
f () { rm -v $1.pdf;  textcleaner -f 50 -o 20 $1.jpg $1.png; djvuize $1.pdf; xpdf $1.pdf }

f () { rm -fv $1.png $1.pdf; djvuize $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 1.0 -f 15" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 1.0 -f 30" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 1.0 -f 45" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 0.5" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 0.25" $1.pdf; xpdf $1.pdf }
f () { cp -fv $1.png $1.pdf       ~/2023.2-C2/
       cp -fv        $1.pdf ~/LATEX/2023-2-C2/
       cat <<%%%
% (find-latexscan-links "C2" "$1")
%%%
}

f 20201213_area_em_funcao_de_theta
f 20201213_area_em_funcao_de_x
f 20201213_area_fatias_pizza



%  __  __       _        
% |  \/  | __ _| | _____ 
% | |\/| |/ _` | |/ / _ \
% | |  | | (_| |   <  __/
% |_|  |_|\__,_|_|\_\___|
%                        
% <make>

 (eepitch-shell)
 (eepitch-kill)
 (eepitch-shell)
# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2023-2-C2-edos-exatas veryclean
make -f 2019.mk STEM=2023-2-C2-edos-exatas pdf

% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2ee"
% ee-tla: "c2m232edosexatas"
% End:
