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Warning: this is an htmlized version!
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% (find-sh0 "cp -v ~/LATEX/2023-2-C2-edos-lineares.pdf /tmp/pen/")
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% file:///home/edrx/LATEX/2023-2-C2-edos-lineares.pdf
% file:///tmp/2023-2-C2-edos-lineares.pdf
% file:///tmp/pen/2023-2-C2-edos-lineares.pdf
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% (find-LATEX "2019.mk")
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% (find-ssr-links "c2m232edols" "2023-2-C2-edos-lineares")
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% (code-eevlinksvideo "c2m232edols" "2023-2-C2-edos-lineares")
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\documentclass[oneside,12pt]{article}
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%
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%\usepackage{proof} % For derivation trees ("%:" lines)
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%\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")
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\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}}
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\catcode`\^^J=10
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\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")
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% «defs-maxima» (to ".defs-maxima")
%L dofile "Maxima2.lua" -- (find-angg "LUA/Maxima2.lua")
\pu
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (c2m232edolsp 1 "title")
% (c2m232edolsa "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo C2 - 2023.2}
\bsk
Aula 35: EDOs lineares
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://anggtwu.net/2023.2-C2.html}
\end{center}
\newpage
% «links» (to ".links")
% (c2m232edolsp 2 "links")
% (c2m232edolsa "links")
{\bf Links}
\scalebox{0.6}{\def\colwidth{16cm}\firstcol{
% (find-books "__analysis/__analysis.el" "stewart-pt" "557" "9.5 Equações Lineares")
% (find-books "__analysis/__analysis.el" "stewart-pt" "561" "9.5 Exercícios")
\par \Ca{StewPtCap9p37} (p.557) 9.5 Equações Lineares
\par \Ca{StewPtCap9p41} (p.561) 9.5 Exercícios
\ssk
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "23" "2.1. Equações lineares")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "29" "Problemas")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "24" "2.1 Linear Differential Equations")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "31" "Problems")
\par \Ca{BoyceDip2p5} (p.23) 2.1 Equações lineares; método dos fatores integrantes
\par \Ca{BoyceDip2p11} (p.29) Problemas
\par \Ca{BoyceDipEng2p4} (p.24) 2.1 Linear Differential Equations; Method of Integrating Factors
\par \Ca{BoyceDipEng2p11} (p.31) Problems
\ssk
% (find-books "__analysis/__analysis.el" "zill-cullen-pt" "68" "2.5. Equações lineares")
% (find-books "__analysis/__analysis.el" "zill-cullen-pt" "77" "2.5. Exercícios")
% (find-books "__analysis/__analysis.el" "zill-cullen" "53" "2.3. Linear equations")
% (find-books "__analysis/__analysis.el" "zill-cullen" "60" "Exercises 2.3")
\par \Ca{ZillCullenCap2p33} (p.68) 2.5 Equações lineares
\par \Ca{ZillCullenCap2p42} (p.77) 2.5 Exercícios
\par \Ca{ZillCullenEngCap2p26} (p.53) 2.3 Linear equations
\par \Ca{ZillCullenEngCap2p33} (p.60) Exercises 2.3
\ssk
% (find-books "__analysis/__analysis.el" "lebl" "40" "1.4 Linear equations and the integrating factor")
% (find-books "__analysis/__analysis.el" "lebl" "43" "1.4.1 Exercises")
\par \Ca{DiffyQsP40} 1.4 Linear equations and the integrating factor
\par \Ca{DiffyQsP43} 1.4.1 Exercises
\msk
}\anothercol{
}}
\newpage
% «defs-bodies» (to ".defs-bodies")
% (c2m232edolsp 3 "defs-bodies")
% (c2m232edolsa "defs-bodies")
\sa {[EL3]}{\CFname{EL}{_3}}
\sa {[S1]}{\CFname{S}{_1}}
\def\P#1{\left( #1 \right)}
\sa{body Stewart}{
\frac{dy}{dx} + P(x)y &=& Q(x) & {[1]} \\
I(x)(y' + P(x)y) &=& (I(x)y)' & {[3]} \\
(I(x)y)' &=& I(x)Q(x) \\
I(x)y &=& \intx{I(x)Q(x)} + C \\
y(x) &=& \frac{1}{I(x)} \left[ \intx{I(x)Q(x)} + C \right] & {[4]} \\
I(x)y' + I(x)P(x)y &=& (I(x)y)' \;=\; I'(x)y + I(x)y' \\
I(x)P(x) &=& I'(x) \\
\int{\frac{1}{I}}\,dI &=& \intx{P(x)} \\
I(x) &=& Ae^{\intx{P(x)}} \\
A &=& \pm e^C \\
A &=& 1 \\
I(x) &=& e^{\intx{P(x)}} & {[5]} \\
}
\sa{body Stewart 2}{
\frac{dy}{dx} + P(x)y &=& Q(x) & {[1]} \\
I(x) &=& e^{\intx{P(x)}} & {[5]} \\
y(x) &=& \frac{1}{I(x)} \left[ \intx{I(x)Q(x)} + C \right] & {[4]} \\
}
\sa{(EL3)}{
\P{\begin{array}{rcl}
f'+fg & = & h \\
G' & = & g \\
f & = & e^{-G}(\intx{e^Gh} + C) \\
\end{array}
}}
\sa{body PQI}{
y' + Py &=& Q \\
I(y' + Py) &=& (Iy)' \\
I(y' + Py) &=& IQ \\
(Iy)' &=& IQ \\
Iy &=& \intx{IQ} \\
y &=& \frac{1}{I} \intx{IQ} \\
I(y' + Py) &=& (Iy)' \\
Iy' + IPy &=& I'y + Iy' \\
IPy &=& I'y \\
IP &=& I' \\
I &=& e^{\intx{P}} \\
}
\sa{body ghm}{
f' + gf &=& h \\
m(f' + gf) &=& (mf)' \\
m(f' + gf) &=& mh \\
(mf)' &=& mh \\
mf &=& \intx{mh} \\
f &=& \frac{1}{m} \intx{mh} \\
m(f' + gf) &=& (mf)' \\
mf' + mgf &=& m'f + mf' \\
mgf &=& m'f \\
mg &=& m' \\
m &=& e^{\intx{g}} \\
&=& e^G \\
}
\sa{body 3}{
f' + gf &=& h \\
G' &=& g \\
f &=& e^{-G} \intx{e^Gh} \\
}
\newpage
% «o-metodo» (to ".o-metodo")
% (c2m232edolsp 3 "o-metodo")
% (c2m232edolsa "o-metodo")
{\bf O método}
\scalebox{0.46}{\def\colwidth{12cm}\firstcol{
Aqui a gente tem a explicação do Stewart de como resolver EDOs
lineares {\sl com todas as partes em português deletadas}:
%
$$\begin{array}{rcll}
\ga{body Stewart}
\end{array}
$$
Repare que sem as partes em português ela vira algo que só gênios
conseguem decifrar -- e um dos nossos objetivos neste curso é aprender
a organizar as contas de modo que elas fiquem fáceis de entender, de
justificar e de verificar.
\msk
Se a gente deixa só as linhas [1], [4] e [5] e põe elas nesta ordem,
%
$$\begin{array}{rcll}
\ga{body Stewart 2}
\end{array}
$$
o {\sl método} fica bem claro: pra resolver uma EDO da forma [1] a
gente define um fator integrante $I(x)$ usando a definição da linha
[5], e aí as nossas soluções vão ser as funções $y(x)$ da linha [4],
onde $C$ é uma constante qualquer.
}\anothercol{
Agora se a gente precisar {\sl resolver} EDOs lineares basta aplicar
um método que cabe em três linhas. Eu prefiro escrever ele usando
outras letras,
%
$$\begin{array}{ccl}
y(x) & ⇒ & f(x) \\
P(x) & ⇒ & g(x) \\
\intx{P(x)} & ⇒ & G(x) \\
Q(x) & ⇒ & h(x) \\
I(x) & ⇒ & m(x) \\
\end{array}
$$
o omitindo os `$(x)$' na maioria dos lugares. A tradução é isto,
%
$$\begin{array}{rcl}
f'+gf & = & h \\
m & = & e^G \\
f & = & \frac{1}{m}(\intx{mh} + C) \\
\end{array}
$$
mas eu vou preferir escrever ela deste jeito:
%
$$\ga{[EL3]} \;=\; \ga{(EL3)}
$$
\bsk
% «exercicio-0» (to ".exercicio-0")
% (c2m232edolsp 3 "exercicio-0")
% (c2m232edolsa "exercicio-0")
{\bf Exercício 0}
O Stewart começa por este exemplo, que ele chama de [2]:
% (find-books "__analysis/__analysis.el" "stewart-pt" "557" "9.5 Equações Lineares")
\par \Ca{StewPtCap9p37} (p.557) $y'+\frac{1}{x}y=2$
Seja $\ga{[S1]} \;=\; \bsm{g := 1/x \\ h:=2 \\ G := \ln x \\ }$.
a) Use $\ga{[EL3]}\ga{[S1]}$ pra obter a solução geral da EDO [2].
b) Chame esta solução geral de $f_1(x)$ -- use um ``seja''! -- e teste-a.
c) Encontre a solução particular que passa pelo ponto $(2,5)$.
d) Chame esta solução particular de $f_2(x)$ -- use um ``seja''! -- e teste-a.
}}
\newpage
{\bf O que realmente importa}
\scalebox{0.7}{\def\colwidth{12cm}\firstcol{
{\bf Exercício importantíssimo!!!}
Entenda isto aqui e reescreva num formato BEM
mais fácil de entender:
%
$$\begin{array}{rcll}
\ga{body Stewart}
\end{array}
$$
}\anothercol{
}}
\newpage
\newpage
% % «metodo-e-formula» (to ".metodo-e-formula")
% % (c2m232edolsp 3 "metodo-e-formula")
% % (c2m232edolsa "metodo-e-formula")
%
% \scalebox{0.6}{\def\colwidth{6cm}\firstcol{
%
% $$\begin{array}[t]{rcl}
% \ga{body PQI}
% \end{array}
% \qquad
% \begin{array}[t]{rcl}
% \ga{body ghm}
% \end{array}
% \qquad
% \begin{array}[t]{rcl}
% \ga{body 3}
% \end{array}
% $$
%
% }\anothercol{
%
% }}
\newpage
% «maxima» (to ".maxima")
% (c2m232edolsp 5 "maxima")
% (c2m232edolsa "maxima")
% (find-es "maxima" "2023-2-edos-lineares")
%M (%i1) e1 : 'diff(y,x) + 1/x * y = 2;
%M (%o1) {\frac{d}{d\,x}}\,y+{\frac{y}{x}}=2
%M (%i2) e2 : ode2(e1,y,x);
%M (%o2) y={\frac{x^2+\mathrm{\%c}}{x}}
%M (%i3) solve(e2, %c);
%M (%o3) \left[ \mathrm{\%c}=x\,y-x^2 \right]
%M (%i4) e3 : solve(e2, %c)[1];
%M (%o4) \mathrm{\%c}=x\,y-x^2
%M (%i5) e4 : subst([x=2,y=5], e2);
%M (%o5) 5={\frac{\mathrm{\%c}+4}{2}}
%M (%i6) solve(e4, %c);
%M (%o6) \left[ \mathrm{\%c}=6 \right]
%L maximahead:sa("stewart exemplo 0 a", "")
\pu
%M (%i7) e4 : solve(e4, %c)[1];
%M (%o7) \mathrm{\%c}=6
%M (%i8) subst(e4,e2);
%M (%o8) y={\frac{x^2+6}{x}}
%M (%i9) define(f2(x), rhs(subst(e4,e2)));
%M (%o9) \mathrm{f2}\left(x\right):={\frac{x^2+6}{x}}
%M (%i10) e5 : subst([y=f2(x)], e1);
%M (%o10) {\frac{d}{d\,x}}\,\left({\frac{x^2+6}{x}}\right)+{\frac{x^2+6}{x^2}}=2
%M (%i11) ev(e5, diff);
%M (%o11) 2=2
%M (%i12)
%L maximahead:sa("stewart exemplo 0 b", "")
\pu
\scalebox{0.6}{\def\colwidth{9cm}\firstcol{
\def\hboxthreewidth {14cm}
\ga{stewart exemplo 0 a}
}\anothercol{
\def\hboxthreewidth {14cm}
\ga{stewart exemplo 0 b}
}}
\GenericWarning{Success:}{Success!!!} % Used by `M-x cv'
\end{document}
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% (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 }
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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-lineares veryclean
make -f 2019.mk STEM=2023-2-C2-edos-lineares pdf
% Local Variables:
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