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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (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{0.6}{\def\colwidth{16cm}\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" "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 \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
\msk
% «links-provas» (to ".links-provas")
% (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}
% «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 \Ca{2jQ97} (2024.2) Quadros das aulas sobre EDOs exatas
\par \Ca{2iQ93} (2024.1) Quadros das aulas sobre EDOs exatas
\par \Ca{2iQ93} (2024.1) Quadros das aulas sobre 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")
% (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}
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