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%  http://anggtwu.net/LATEX/2024-1-C2-coefs-a-determinar.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-cps   "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-anggs "Caepro5 Piecewise2 Maxima2")
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\catcode`\^^J=10
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% «defs-T-and-B»  (to ".defs-T-and-B")
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% «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
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%  _____ _ _   _                               
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\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 2 - 2024.1}

\bsk

Aula 35: o método dos

coeficientes a determinar

\bsk

Eduardo Ochs - RCN/PURO/UFF

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

\end{center}

\newpage

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{\bf Links}

\scalebox{0.6}{\def\colwidth{18cm}\firstcol{

% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "134" "3.5. Equações não-homogêneas; método dos coeficientes indeterminados")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "176" "4.2. Equações homogêneas com coeficientes constantes")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "183" "4.3. O método dos coeficientes indeterminados")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "133" "3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "174" "4.2 Homogeneous Differential Equations with Constant Coefficients")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "181" "4.3 The Method of Undetermined Coefficients")
\par \Ca{BoyceDip3p33} (p.134) 3.5. Equações não-homogêneas; método dos coeficientes indeterminados
\par \Ca{BoyceDip4p9} (p.176) 4.2. Equações homogêneas com coeficientes constantes
\par \Ca{BoyceDip4p16} (p.183) 4.3. O método dos coeficientes indeterminados
\par \Ca{BoyceDipEng3p34} (p.133) "3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
\par \Ca{BoyceDipEng4p9} (p.174) 4.2 Homogeneous Differential Equations with Constant Coefficients
\par \Ca{BoyceDipEng4p16} (p.181) 4.3 The Method of Undetermined Coefficients

\msk

% (find-books "__analysis/__analysis.el" "zill-cullen-pt" "182" "4.4. Coeficientes indeterminados - abordagem por superposição")
% (find-books "__analysis/__analysis.el" "zill-cullen-pt" "195" "4.5. Operadores diferenciais")
% (find-books "__analysis/__analysis.el" "zill-cullen-pt" "196" "Exemplo 1: ...pode ser fatorado...")
% (find-books "__analysis/__analysis.el" "zill-cullen-pt" "201" "4.6. Coeficientes indeterminados - abordagem por anuladores")
% (find-books "__analysis/__analysis.el" "zill-cullen" "140" "4.4 Undetermined Coefficients--Superposition Approach")
% (find-books "__analysis/__analysis.el" "zill-cullen" "150" "4.5 Undetermined Coefficients--Annihilator Approach")
% (find-books "__analysis/__analysis.el" "zill-cullen" "150" "FACTORING OPERATORS")
\par \Ca{ZillCullenCap4p46} (p.182) 4.4. Coeficientes indeterminados - abordagem por superposição
\par \Ca{ZillCullenCap4p59} (p.195) 4.5. Operadores diferenciais
\par \Ca{ZillCullenCap4p60} (p.196) Exemplo 1: ...pode ser fatorado...
\par \Ca{ZillCullenCap4p65} (p.201) 4.6. Coeficientes indeterminados - abordagem por anuladores
\par \Ca{ZillCullenEngCap4p30} (p.140) 4.4 Undetermined Coefficients - Superposition Approach
\par \Ca{ZillCullenEngCap4p40} (p.150) 4.5 Undetermined Coefficients - Annihilator Approach
\par \Ca{ZillCullenEngCap4p40} (p.150) Factoring operators

% (find-books "__analysis/__analysis.el" "trench" "229" "5.4 The Method of Undetermined Coefficients I")
% (find-books "__analysis/__analysis.el" "trench" "238" "5.5 The Method of Undetermined Coefficients II")
% (find-books "__analysis/__analysis.el" "trench" "497" "9.4 Variation of Parameters for Higher Order Equations")

\msk

% (find-angg ".emacs" "c2q232" "coeficientes a determinar")
\par \Ca{2hQ85} Aula 38 de 2023.2, sobre coeficientes a determinar (21/nov/2023)

}\anothercol{
}}

\newpage

% «questao-2»  (to ".questao-2")
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% (c2m231p2a   "questao-2")

{\bf Questão 2}

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

\vspace*{-0.4cm}

\T(Total: 4.0 pts)

Lembre que nós vimos dois tipos de EDOs lineares com coeficientes
constantes --- ``EDOLCCs'' --- no curso: o primeiro tipo tinha
soluções básicas da forma $e^{ax}$ e $e^{bx}$, onde $a$ e $b$ são
reais, e o segundo tipo tinha ``soluções básicas complexas'' da forma
$e^{(a+ib)x}$ e $e^{(a-ib)x}$ e ``soluções básicas reais'' da forma
$e^{αx}\cos βx$ e $e^{αx}\sen βx$; as soluções básicas reais eram
combinações lineares das soluções básicas complexas e vice-versa.

\msk

Sejam $(**)$ e $({*}{*}{*})$ as EDOs abaixo:
%
$$\begin{array}{rcll}
  y'' +  y' - 20y &=& 0 & \qquad (**) \\
  y'' + 4y' + 29y &=& 0 & \qquad ({*}{*}{*}) \\
  \end{array}
$$

A EDO $(**)$ é do primeiro tipo e a EDO $({*}{*}{*})$ é do segundo tipo.

\bsk

\standout{Original aqui:}

\par \Ca{2gT135} (2023.1) P2, questão 2

\vspace*{-2cm}

}\anothercol{

{}

a) \B (0.5 pts) Encontre as soluções básicas e a solução geral da EDO
$(**)$. Dê um nome para cada uma delas.

\msk

b) \B (1.5 pts) Encontre uma solução da EDO $(**)$ -- vou chamá-la de
$g(x)$ -- que obedece $g(0) = 4$ e $g'(0)=5$, e teste-a. Dica: você
vai ter que resolver um sistema pra descobrir a quantidade certa de
cada ``vetor'' na combinação linear!

\bsk

c) \B (0.5 pts) Diga quais são as ``soluções básicas complexas'' e as
``soluções básicas reais'' para a EDO $({*}{*}{*})$.

\msk

d) \B (1.5 pts) Escolha uma das suas ``soluções básicas reais'' do
item anterior e verifique que ela realmente é uma solução da EDO
$({*}{*}{*})$.

}}


\newpage

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{\bf Três exemplos}

\scalebox{0.6}{\def\colwidth{10.5cm}\firstcol{

Na aula 38 -- fotos do quadros: \Ca{2hQ85} -- nós resolvemos três
exemplos:
%
% (find-angg ".emacs" "c2q232" "coeficientes a determinar")
%\par \Ca{2hQ85} Aula 38 de 2023.2, sobre coeficientes a determinar
%
$$\begin{array}{cl}
 \text{(Ex1):} & y''-3y'-4y = 3e^{2x} \\
 \text{(Ex2):} & y''-3y'-4y = 2 \sen x \\
 \text{(Ex2.5):} & y''-3y'-4y = 4x^2-1 \\
 \end{array}
$$

% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "134" "3.5. Equações não-homogêneas; método dos coeficientes indeterminados")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "136" "Exemplo 1")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "133" "3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "135" "Example 1")

Eu peguei esses exemplos daqui:

\par \Ca{BoyceDip3p33} 3.5 (...) coeficientes indeterminados
\par \Ca{BoyceDip3p35} Exemplo 1
\par \Ca{BoyceDipEng3p34} 3.5 (...) Undetermined Coefficients
\par \Ca{BoyceDipEng3p36} Example 1

}\anothercol{
}}


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