% (find-LATEX "2025-1-C2-edos-exatas.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2025-1-C2-edos-exatas.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2025-1-C2-edos-exatas.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2025-1-C2-edos-exatas.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2025-1-C2-edos-exatas.pdf"))
% (defun e () (interactive) (find-LATEX "2025-1-C2-edos-exatas.tex"))
% (defun o () (interactive) (find-LATEX "2024-2-C2-edos-exatas.tex"))
% (defun u () (interactive) (find-latex-upload-links "2025-1-C2-edos-exatas"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2025-1-C2-edos-exatas.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (defun oe () (interactive) (find-2a '(o) '(e)))
%          (code-eec-LATEX "2025-1-C2-edos-exatas")
% (find-pdf-page   "~/LATEX/2025-1-C2-edos-exatas.pdf")
% (find-sh0 "cp -v  ~/LATEX/2025-1-C2-edos-exatas.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2025-1-C2-edos-exatas.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2025-1-C2-edos-exatas.pdf")
%   file:///home/edrx/LATEX/2025-1-C2-edos-exatas.pdf
%               file:///tmp/2025-1-C2-edos-exatas.pdf
%           file:///tmp/pen/2025-1-C2-edos-exatas.pdf
%  http://anggtwu.net/LATEX/2025-1-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 "2025-1-C2-edos-exatas" "2" "c2m251edosexatas" "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-V»		(to "defs-V")
% «.title»		(to "title")
% «.links»		(to "links")
% «.links-stewart»		(to "links-stewart")
% «.links-boyce»		(to "links-boyce")
% «.links-zillcullen»		(to "links-zillcullen")
% «.links-diffyqs»		(to "links-diffyqs")
% «.links-provas»		(to "links-provas")
% «.links-quadros»		(to "links-quadros")
% «.metodo-e-exemplo»		(to "metodo-e-exemplo")
% «.uma-questao-de-prova»	(to "uma-questao-de-prova")
% «.maxima-exact»		(to "maxima-exact")
% «.maxima-caixinhas-1»		(to "maxima-caixinhas-1")
% «.maxima-caixinhas-2»		(to "maxima-caixinhas-2")



\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-LATEX "dednat7-test1.tex")
%\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/2025-1-C2.pdf}
\def\drafturl{http://anggtwu.net/2025.1-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 "dednat7load.lua"}  % (find-LATEX "dednat7load.lua")
\directlua{dednat7preamble()}         % (find-angg "LUA/DednatPreamble1.lua")
\directlua{dednat7oldheads()}         % (find-angg "LUA/Dednat7oldheads.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

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

\sa{reset-S2}{
  \sa {z}   {(x^2+y^2)}
  \sa {z_x} {(2x)}
  \sa {z_y} {(2y)}
  }



%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (c2m251edosexatasp 1 "title")
% (c2m251edosexatasa   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 2 - 2025.1}

\bsk

Aulas 46 e 47: EDOs exatas

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2025.1-C2.html}

\end{center}

\newpage

% «links»  (to ".links")
% (c2m251edosexatasp 2 "links")
% (c2m251edosexatasa   "links")

{\bf Links}

\scalebox{0.57}{\def\colwidth{15.5cm}\firstcol{

% «links-stewart»  (to ".links-stewart")
% (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

% «links-boyce»  (to ".links-boyce")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "72" "2.6. Equações Exatas")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "73" "Começamos integrando")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "74" "Exemplo 2 ... Integrando")
% (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{BoyceDip2p55} (p.73) Começamos integrando
\par \Ca{BoyceDip2p56} (p.74) Exemplo 2 ... Integrando
\par \Ca{BoyceDip2p58} (p.76) Problemas \standout{$←$ façam os exercícios 1 a 4}
\par \Ca{BoyceDipEng2p50} (p.70) 2.6 Exact Differential Equations and Integrating Factors
\par \Ca{BoyceDipEng2p55} (p.75) Problems

\ssk

% «links-zillcullen»  (to ".links-zillcullen")
% (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

% «links-diffyqs»  (to ".links-diffyqs")
% (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

}\anothercol{

% «links-provas»  (to ".links-provas")
% (c2m222edoexsp 2 "links")
% (c2m222edoexsa   "links")
% (c2m192p2p)
% (c2m192p2a)
% 2jT273: (c2m241p2p 4 "questao-4")
%         (c2m241p2a   "questao-4")
% 2iT227: (c2m241p2p 4 "questao-4")
%         (c2m241p2a   "questao-4")
%  2yT14: (c2m192p2p 2 "questao-4")
%         (c2m192p2a   "questao-4")
\par Provas:
\par \Ca{2jT273} (2024.2) P2
\par \Ca{2iT227} (2024.1) P2
\par \Ca{2yT14} (2019.2) P2

\msk

% «links-quadros»  (to ".links-quadros")
% (find-angg ".emacs" "c2q242" "07/01: EDOs exatas")
% (find-angg ".emacs" "c2q241" "EDOs exatas")
% (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 Quadros:
\par \Ca{2jQ97} (2024.2)
\par \Ca{2iQ93} (2024.1)
\par \Ca{2iQ93} (2024.1)
\par \Ca{2hQ73} (2023.2)
\par \Ca{2yQ106} (2019.2)



}}

\newpage

% «metodo-e-exemplo»  (to ".metodo-e-exemplo")
% (c2m251edosexatasp 3 "metodo-e-exemplo")
% (c2m251edosexatasa   "metodo-e-exemplo")
% (c2m242edosexatasp 3 "metodo-e-exemplo")
% (c2m242edosexatasa   "metodo-e-exemplo")
% (c2m241edosexatasp 3 "metodo-e-exemplo")
% (c2m241edosexatasa   "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)}  \\ \\[-5pt]
  \ga{[S2]} &=& \ga{reset-S2}
                \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}
$

\bsk
\bsk

$\ga{reset-S2}
  \begin{array}[t]{rcl}
    \ga{[E5]}\ga{[S2]} &=& \ga{(E5)}  \\ \\[-5pt]
    \ga{[E3]}\ga{[S2]} &=& \ga{(E3)} \\ \\[-5pt]
    \ga{[E2]}\ga{[S2]} &=& \ga{(E2)} \\
  \end{array}
$


}}


\newpage

% «uma-questao-de-prova»  (to ".uma-questao-de-prova")
% (c2m241edosexatasp 5 "uma-questao-de-prova")
% (c2m241edosexatasa   "uma-questao-de-prova")
% (c2m232edosexatasp 5 "uma-questao-de-prova")
% (c2m232edosexatasa   "uma-questao-de-prova")
% (find-es "maxima" "2024.1-exact")

{\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}

}\anothercol{
}}

\newpage

% «maxima-exact»  (to ".maxima-exact")
% (c2m251edosexatasp 5 "maxima-exact")
% (c2m251edosexatasa   "maxima-exact")
% (find-angg "MAXIMA/2025-1-exact.mac")

%M (%i1) load("2025-1-exact.mac")$
%M (%i2) ee : Exact_from0()$
%M (%i3) ee@@E5();
%M (%o3) \begin{pmatrix}d\,z&\mbox{ = }&z_x \,\mathrm{dx}+z_y \,\mathrm{dy}&\mbox{ = }&0\cr {\frac{d}{d\,x}}\,z&\mbox{ = }&z_x +z_y \,\left({\frac{d}{d\,x}}\,y\right)&\mbox{ = }&0\cr z&\mbox{ = }&C&&\cr \end{pmatrix}
%M (%i4) ee : Exact_from1(x^2+y^2)$
%M (%i5) ee@@E5();
%M (%o5) \begin{pmatrix}d\,\left(y^2+x^2\right)&\mbox{ = }&2\,x\,\mathrm{dx}+2\,y\,\mathrm{dy}&\mbox{ = }&0\cr {\frac{d}{d\,x}}\,\left(y^2+x^2\right)&\mbox{ = }&2\,x+2\,y\,\left({\frac{d}{d\,x}}\,y\right)&\mbox{ = }&0\cr y^2+x^2&\mbox{ = }&C&&\cr \end{pmatrix}
%M (%i6) ee : Exact_from2(2*x, 2*y)$
%M (%i7) ee@@E5();
%M (%o7) \begin{pmatrix}d\,z&\mbox{ = }&2\,x\,\mathrm{dx}+2\,y\,\mathrm{dy}&\mbox{ = }&0\cr {\frac{d}{d\,x}}\,z&\mbox{ = }&2\,x+2\,y\,\left({\frac{d}{d\,x}}\,y\right)&\mbox{ = }&0\cr z&\mbox{ = }&C&&\cr \end{pmatrix}
%L maximahead:sa("Exact", "")
\pu

%M (%i8) ee@@ode();
%M (%o8) 2\,y\,\left({\frac{d}{d\,x}}\,y\right)+2\,x=0
%M (%i9) ee@@odesolve();
%M (%o9) -\left({\frac{y^2}{2}}\right)={\frac{x^2}{2}}+\mathrm{\%c}
%M (%i10) sols  : ee@@odesols();
%M (%o10) \left[ y=-\sqrt{-x^2-2\,\mathrm{\%c}} , y=\sqrt{-x^2-2\,\mathrm{\%c}} \right] 
%M (%i11) sol   : sols[1];
%M (%o11) y=-\sqrt{-x^2-2\,\mathrm{\%c}}
%M (%i12) test1 : ee@@odesubst(sol);
%M (%o12) 2\,x-2\,\sqrt{-x^2-2\,\mathrm{\%c}}\,\left({\frac{d}{d\,x}}\,\left(-\sqrt{-x^2-2\,\mathrm{\%c}}\right)\right)=0
%M (%i13) test2 : ev(test1, diff);
%M (%o13) 0=0
%L maximahead:sa("Exact 2", "")
\pu

\scalebox{0.55}{\def\colwidth{10cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {10cm}
\ga{Exact}

}\anothercol{

\vspace*{0cm}
\def\hboxthreewidth {10cm}
\ga{Exact 2}

}}


\newpage

% «maxima-caixinhas-1»  (to ".maxima-caixinhas-1")
% (c2m251edosexatasp 6 "maxima-caixinhas-1")
% (c2m251edosexatasa   "maxima-caixinhas-1")

%M (%i1) load("2025-1-exact.mac")$
%M (%i2) ee : Exact_from1(x^2*y^3);
%M (%o2) \mathrm{Exact}\left(x^2\,y^3 , 2\,x\,y^3 , 3\,x^2\,y^2\right)
%M (%i3) ee@@caixinhas_3();
%M (%o3) \left[ \begin{pmatrix}0&0&1&0\cr 0&0&0&0\cr 0&0&0&0\cr 0&0&0&0\cr \end{pmatrix} , \begin{pmatrix}0&2&0&0\cr 0&0&0&0\cr 0&0&0&0\cr 0&0&0&0\cr \end{pmatrix} , \begin{pmatrix}0&0&0&0\cr 0&0&3&0\cr 0&0&0&0\cr 0&0&0&0\cr \end{pmatrix} \right] 
%M (%i4) ee@@caixinhas_2();
%M (%o4) \left[ \begin{pmatrix}0&2&0&0\cr 0&0&0&0\cr 0&0&0&0\cr 0&0&0&0\cr \end{pmatrix} , \begin{pmatrix}0&0&0&0\cr 0&0&3&0\cr 0&0&0&0\cr 0&0&0&0\cr \end{pmatrix} \right] 
%M (%i5) ee : Exact_from1(x^2+y^2);
%M (%o5) \mathrm{Exact}\left(y^2+x^2 , 2\,x , 2\,y\right)
%M (%i6) ee@@caixinhas_3();
%M (%o6) \left[ \begin{pmatrix}0&0&0&0\cr 1&0&0&0\cr 0&0&0&0\cr 0&0&1&0\cr \end{pmatrix} , \begin{pmatrix}0&0&0&0\cr 0&0&0&0\cr 0&0&0&0\cr 0&2&0&0\cr \end{pmatrix} , \begin{pmatrix}0&0&0&0\cr 0&0&0&0\cr 2&0&0&0\cr 0&0&0&0\cr \end{pmatrix} \right] 
%M (%i7) ee@@caixinhas_2();
%M (%o7) \left[ \begin{pmatrix}0&0&0&0\cr 0&0&0&0\cr 0&0&0&0\cr 0&2&0&0\cr \end{pmatrix} , \begin{pmatrix}0&0&0&0\cr 0&0&0&0\cr 2&0&0&0\cr 0&0&0&0\cr \end{pmatrix} \right] 
%L maximahead:sa("caixinhas", "")
\pu

\scalebox{0.45}{\def\colwidth{9cm}\firstcol{

\vspace*{-0.5cm}
\def\hboxthreewidth {14cm}
\ga{caixinhas}

}\anothercol{
}}

\newpage

% «maxima-caixinhas-2»  (to ".maxima-caixinhas-2")
% (c2m251edosexatasp 7 "maxima-caixinhas-2")
% (c2m251edosexatasa   "maxima-caixinhas-2")

%M (%i1) [maxxp,maxyp] : [2,2];
%M (%o1) \left[ 2 , 2 \right] 
%M (%i2) "1"$  ee : Exact_from3("?", 2*x+3, 2*y-2);
%M (%o3) \mathrm{Exact}\left(\mbox{ ? } , 2\,x+3 , 2\,y-2\right)
%M (%i4) ee@@caixinhas_2();
%M (%o4) \left[ \begin{pmatrix}0&0&0\cr 0&0&0\cr 3&2&0\cr \end{pmatrix} , \begin{pmatrix}0&0&0\cr 2&0&0\cr -2&0&0\cr \end{pmatrix} \right] 
%M (%i5) ee@@effsc();
%M (%o5) \left[ \begin{pmatrix}0&0&0\cr 0&0&0\cr 0&0&0\cr \end{pmatrix}=\begin{pmatrix}0&0&0\cr 0&0&0\cr 0&0&0\cr \end{pmatrix} , \begin{pmatrix}\mbox{ ? }&0&0\cr \mbox{ ? }&0&0\cr \mbox{ ? }&3&1\cr \end{pmatrix} , \begin{pmatrix}1&0&0\cr -2&0&0\cr \mbox{ ? }&\mbox{ ? }&\mbox{ ? }\cr \end{pmatrix} \right] 
%M (%i6) "2"$  ee : Exact_from3("?", 2*x+4*y, 2*x-2*y);
%M (%o7) \mathrm{Exact}\left(\mbox{ ? } , 4\,y+2\,x , 2\,x-2\,y\right)
%M (%i8) ee@@caixinhas_2();
%M (%o8) \left[ \begin{pmatrix}0&0&0\cr 4&0&0\cr 0&2&0\cr \end{pmatrix} , \begin{pmatrix}0&0&0\cr -2&0&0\cr 0&2&0\cr \end{pmatrix} \right] 
%M (%i9) ee@@effsc();
%M (%o9) \left[ \begin{pmatrix}0&0&0\cr 0&0&0\cr 4&0&0\cr \end{pmatrix}=\begin{pmatrix}0&0&0\cr 0&0&0\cr 2&0&0\cr \end{pmatrix} , \begin{pmatrix}\mbox{ ? }&0&0\cr \mbox{ ? }&4&0\cr \mbox{ ? }&0&1\cr \end{pmatrix} , \begin{pmatrix}-1&0&0\cr 0&2&0\cr \mbox{ ? }&\mbox{ ? }&\mbox{ ? }\cr \end{pmatrix} \right] 
%L maximahead:sa("caixinhas?", "")
\pu

%M (%i10) [maxxp,maxyp] : [3,3];
%M (%o10) \left[ 3 , 3 \right] 
%M (%i11) "3"$  ee : Exact_from3("?", 3*x^2 - 2*x*y + 2, 6*y^2 - x^2 + 3);
%M (%o12) \mathrm{Exact}\left(\mbox{ ? } , -\left(2\,x\,y\right)+3\,x^2+2 , 6\,y^2-x^2+3\right)
%M (%i13) ee@@caixinhas_2();
%M (%o13) \left[ \begin{pmatrix}0&0&0&0\cr 0&0&0&0\cr 0&-2&0&0\cr 2&0&3&0\cr \end{pmatrix} , \begin{pmatrix}0&0&0&0\cr 6&0&0&0\cr 0&0&0&0\cr 3&0&-1&0\cr \end{pmatrix} \right] 
%M (%i14) ee@@effsc();
%M (%o14) \left[ \begin{pmatrix}0&0&0&0\cr 0&0&0&0\cr 0&0&0&0\cr 0&-2&0&0\cr \end{pmatrix}=\begin{pmatrix}0&0&0&0\cr 0&0&0&0\cr 0&0&0&0\cr 0&-2&0&0\cr \end{pmatrix} , \begin{pmatrix}\mbox{ ? }&0&0&0\cr \mbox{ ? }&0&0&0\cr \mbox{ ? }&0&-1&0\cr \mbox{ ? }&2&0&1\cr \end{pmatrix} , \begin{pmatrix}2&0&0&0\cr 0&0&0&0\cr 3&0&-1&0\cr \mbox{ ? }&\mbox{ ? }&\mbox{ ? }&\mbox{ ? }\cr \end{pmatrix} \right] 
%L maximahead:sa("caixinhas? 2", "")
\pu


\scalebox{0.4}{\def\colwidth{13cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{caixinhas?}

}\anothercol{

\vspace*{0cm}
\def\hboxthreewidth {16cm}
\ga{caixinhas? 2}

}}




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

\end{document}

% (find-pdfpages2-links "~/LATEX/" "2025-1-C2-edos-exatas")


% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2ee"
% ee-tla: "c2m251edosexatas"
% End:
