% (find-angg "LATEX/2009jun05.tex")
% (find-dn4ex "edrx08.sty")
% (find-angg ".emacs.templates" "s2008a")
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2009jun05.tex && latex    2009jun05.tex"))
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2009jun05.tex && pdflatex 2009jun05.tex"))
% (eev "cd ~/LATEX/ && Scp 2009jun05.{dvi,pdf} edrx@angg.twu.net:slow_html/LATEX/")
% (defun d () (interactive) (find-dvipage "~/LATEX/2009jun05.dvi"))
% (find-dvipage "~/LATEX/2009jun05.dvi")
% (find-pspage  "~/LATEX/2009jun05.pdf")
% (find-pspage  "~/LATEX/2009jun05.ps")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o 2009jun05.ps 2009jun05.dvi")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 600 -P pk -o 2009jun05.ps 2009jun05.dvi && ps2pdf 2009jun05.ps 2009jun05.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o tmp.ps tmp.dvi")
% (find-pspage  "~/LATEX/tmp.ps")
% (ee-cp "~/LATEX/2009jun05.pdf" (ee-twupfile "LATEX/2009jun05.pdf") 'over)
% (ee-cp "~/LATEX/2009jun05.pdf" (ee-twusfile "LATEX/2009jun05.pdf") 'over)

\documentclass[oneside]{book}
\usepackage[latin1]{inputenc}
\usepackage{edrx08}       % (find-dn4ex "edrx08.sty")
%L process "edrx08.sty"  -- (find-dn4ex "edrx08.sty")
\input edrxheadfoot.tex   % (find-dn4ex "edrxheadfoot.tex")
\begin{document}

\input 2009jun05.dnt

%
% (eedn4-51-bounded)

%Index of the slides:
%\msk
% To update the list of slides uncomment this line:
%\makelos{tmp.los}
% then rerun LaTeX on this file, and insert the contents of "tmp.los"
% below, by hand (i.e., with "insert-file"):
% (find-fline "tmp.los")
% (insert-file "tmp.los")

\def\ddx{\frac{d}{dx}}
\def\dydx{\frac{dy}{dx}}
\def\ddth{\frac{d}{d\theta}}
\def\sen{\operatorname{sen}}
\def\sec{\operatorname{sec}}
\def\ln{\operatorname{ln}}

Notas sobre equações diferenciais de 1ª ordem

EDOs da forma $y'=\aa y$:
%:
%:    y'=\aa"y
%:    --------
%:    y'/y=\aa       
%:  ------------     
%:  (\ln"y)'=\aa     y=\expÅ\aa		 
%:  ------------     --------------------
%:   \ln"y=Å\aa      y'=(\expÅ\aa)(Å\aa)'
%:   ----------      --------------------
%:   y=\expÅ\aa      y'=y\aa             
%:
%:     ^foo1         ^foo2
%:
$$\ded{foo1} \qquad \ded{foo2}$$

\msk

EDOs da forma $y'+\aa y=\bb$:
%:
%:                  f=\expÅ\aa
%:  -------------   ----------
%:  (fy)'=fy'+f'y    f'=f\aa      y'+\aa"y=\bb
%:  ------------------------     ---------------
%:      (fy)'=fy'+f\aa"y         fy'+f\aa"y=f\bb
%:      ----------------------------------------
%:                (fy)'=f\bb
%:                ----------
%:                 fy=Åf\bb
%:                ----------
%:                y=(Åf\bb)/f
%:
%:                 ^foo3
%:
$$\ded{foo3}$$

\msk

EDOs da forma $y'=f(x)/g(y)$:

Se $u=u(x)$, $v=v(y)$ e $dy/dx = u_x/v_y$ então:

$v_y\,dy = u_x\,dx$

$dv = v_y\,dy = u_x\,dx = du$

$\frac{dv}{du}=1$

As soluções são da forma $v-u = \text{constante}$, isto é,

as curvas de nível de $v(y)-u(x)$.


\msk

EDOs separáveis da forma $A(x,y)+B(x,y)y'=0$, onde $A_y=B_x$:

Se $\psi=\psi(x,y)$ então as soluções de $\psi_x+\psi_y y'=0$ são as
curvas de nível de $\psi$.

Se $A=A(x,y)$, $B=B(x,y)$ e $A_y=B_x$ então as soluções de $A+By'=0$
são as curvas de nível de $\psi$, onde:
%
$$\begin{array}{rcl}
  \psi(x_1,y_1) &=& Å_{x_0}^{x_1} A(x,y_0)\,dx + Å_{y_0}^{y_1} B(x_1,y)\,dy \\
                &=& Å_{y_0}^{y_1} B(x_0,y)\,dy + Å_{x_0}^{x_1} A(x,y_1)\,dy \\
  \end{array}
$$



%

\end{document}

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