From 51a00b568b817957a11536fbaf950c33182f8fa1 Mon Sep 17 00:00:00 2001 From: Charlie Root Date: Sat, 28 Mar 2026 22:31:28 -0300 Subject: [PATCH] Differential equations. --- calculus.pdf | Bin 911913 -> 922081 bytes calculus.tex | 156 ++++++++++++++++++++++++++++++++++++++++++++++++++- 2 files changed, 155 insertions(+), 1 deletion(-) diff --git a/calculus.pdf b/calculus.pdf index b3c9cf7fcb2572f38058b0e31ad2f7c36d803dfd..6717db758ab45a149bc04e7ca5eea8a8d9c5326a 100644 GIT binary patch delta 164701 zcmYJ4b95!q60dV&I};ld+vdcY*tYGQIGNZd6HLsBZQHhOn{V#DZ@srx|FOHey1Kge z>b<{R{p-?suAO}DYDX9lpbiPME(E3#5x@otbVtIdQ@~_G0B|HZ!F;PrCT>Iod|LGCV6vV)*-XlLIb$|=NZct1()=qK!?N)&m?DNZ{cc5!ph2;gvLSx9MG0cTH{3T zJk!`t-3JF)^8eK>bxVApQj=aZ<1012;IU-C*G$26BC~3L{CxdL&!ZNhpq!LyL?I)x zW$*OQ!4I%ZMw7ugfa1Xv!4d1VbZv?Fi6B#ALs#D8a}ja}ZHCDhsQ!J2;W!3OupccB z32;9cRnm+|mVh0WEG3c;oEJTV9W)I!XN0b^?sXJH!-Dyf9MaeWZxS=DP?*qzxTf<_b4Vtm;19y)-Phy+52U;ld) z(Km9a%P?g2U>K@0cFar?V}8;+bP-s&eNB;>oZW|EFn9?mLo#-JAcO|E>&R&T6fKnk 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Acima, nós fomos até $2^3$ no denominador, portanto temos $2^4$ termos, que são $2^7$ metades e a unidade com a qual a série se inicia. Isso então significa... +\section{Equações diferenciais} + +Equações diferenciais é certamente a aplicação mais importante de +Cálculo porque quase qualquer coisa na vida envolve uma quantidade que +varia com o tempo e em seguida deseja-se por uma fórmula que expresse +a acumulação dessa coisa que varia. Embora não sejamos obcecados por +equações diferenciais, gostaríamos de documentar as mais simples +observações sobre o tema. + +Uma equação diferencial é chamada de {\em ordinária} se a função que +soluciona a equação depende de apenas uma única váriavel independente, +ou seja, a função desejada é $y(x)$. Se ela depender de mais de uma +variável, então trata-se de uma equação diferencial parcial, quando a +solução é uma função como, por exemplo, $y(x_1, x_2)$. + +Uma equação é chamada de {\em separável de primeira ordem} quando ela +pode ser escrita como +% + $$\frac{dy}{dx} = f(x) g(y)$$ +% +porque assim é possível separar $y$ e $x$ em cada lado da equação e +simplesmente integrar ambos os lados da equação usando-se uma +aplicação do seguinte teorema. + +\masciith{} \label{th1} Declare $y = Y(x)$ uma solução de +% +\begin{align} \label{8.45} A(y) y' = Q(x) \end{align} +% +tal que $Y'$ é contínua num intervalo $I$. Assuma que $Q$ and $(A +\circ Y)$ sejam contínuas em $I$. Declare $G$ uma antiderivada de +$A$. Daí $y$ satisfaz +% +\begin{align}\label{8.46}G(y) = \int Q(x) \dx + C.\end{align} +% +Conversamente, se $y$ satisfaz (\ref{8.46}) então $y$ é solução de +(\ref{8.45}). \qed \theend + +\masciiproof Já que $Y$ é solução de (\ref{8.45}), então +% +\begin{align} \label{8.47} A(Y(x))Y'(x) = Q(x) \end{align} +% +pra cada $x$ em $I$. Já que $G'= A$, então (\ref{8.45}) nos dá +% + $$G'(Y(x))Y'(x) = Q(x).$$ +% +Mas $G'(Y(x))Y'(x)$ é a derivada de $G(Y(x))$, que então é uma +antiderivada de $Q$, isto é +% + \begin{align} \label{8.48} G(Y(x)) = \int Q(x) \dx + C,\end{align} +% +e assim a primeira parte do teorema está provada. (Note a constante +C. Quando integramos $G'(Y(x))Y'(x)$, uma constante aparece. Chame-a +de D e declare $C = -D$.) + +Agora assuma que $y$ satisfaz (\ref{8.46}). Derivando-a, obtemos +% + $$ A(y)y' = Q(x),$$ +% +que era o que desejávamos.\profend + +Temos aqui apenas uma relação entre equações. A mesma função $y$ que +satisfaz $A(y) y' = Q(x)$ satisfaz também $$G(y) = \int Q(x) \dx + +C.$$ + +\ghostsection{A importância dessa relação}. Considere a equação +$y'(x) = y(x)$, onde $y'$ é a derivada de $y$. Implícito na notação +está o fato de que $y$ é uma variável dependente de $x$. Álgebra nos +permite dizer que $$(1/y(x)) y'(x) = 1.$$ + +Declare $A(y) = 1/y$ e $Q(x) = 1$. Declare $G(y)$ uma +anti-derivada de $A(y)$ --- como por exemplo $\ln|y|$. O teorema acima nos +permite dizer que a função $y$ também satisfaz a equação +$$G(y) = \int Q(x) \dx + C.$$ Logo, para descobrir uma anti-derivada +de $A(y)$, basta integrar $Q(x)$ com respeito a $x$. Perceba que isto +é o mesmo que integrar $A(y)$ com respeito a $y$, ou integrar $A(y) +y'(x)$ com respeito a $x$. Escolhamos então a integração mais fácil e +de dessa forma obtemos a das outras também. + +Esse é o raciocínio necessário para solucionar equações diferenciais +separáveis de primeira ordem. O que alguns livros-texto fazem, como +por exemplo o famoso ``Calculus'' de James Stewart, é aplicar o +teorema através de um truque da notação de Leibniz. + +Olhemos de mais perto. Considere novamente $dy/dx = y$. Para +solucionar essa equação, precisamos invocar o teorema acima. Em vez +de invocá-lo, entretanto, suponhamos que $dy$ e $dx$ sejam, em si, +variáveis distintas e por mera questão de investigação, operemos a +equação da seguinte forma: +\begin{align} + \dy/\dx &= y\\ + (1/y)\dy &= \dx +\end{align} +Voltemos agora nossa atenção à equação +\begin{align} + \int (1/y)\dy = \int dx. +\end{align} +Sabemos que o lado direito é equivalente a $x + C_1$ e que o lado +esquerdo é equivalente a $\ln|y| + C_2$. Podemos isolar $y$ através +de uma aplicação da inversa de $\ln$ e estaríamos satisfeitos, ou +seja, se houver uma conexão entre (5) e (7), temos um procedimento que +é fácil de lembrar --- devido a algebrização dos símbolos $\dy$ e +$\dx$ --- e que nos permite expressar $y$ em função de $x$ --- isto é, +solucionar a equação. Felizmente a conexão existe como vimos acima. + +\subsection{O modelo logístico} + +Uma equação diferencial famosa é o modelo logístico, que modela +crescimento populacional. É chamada de ``lei do crescimento natural'' +o fato de que a velocidade de crescimento de uma população é +proporcional ao tamanho atual dela, o que é bem sensato. Em símbolos, +essa lei se traduz em +% + $$\fr{dP}{dt} = kP,$$ +% +sendo $k$ alguma constante específica relativa à população em si. Por +exemplo, uma população de bactérias pode ter uma certa constante e uma +população de búfalos, outra. + +Essa equação por si só já é um modelo de crescimento populacional, mas +é muito simplista: ela assume, por exemplo, que a população pode +crescer pra sempre, o que até nem é absurdo quando a população está em +início de expansão, mas se torna absurda quando a população começa a +passar de sua ``capacidade máxima de suporte'', que vamos chamar de +$M$. + +O modelo logístico leva essa capacidade máxima em consideração e faz +com que o crescimento se torne decaimento se a população estiver acima +dessa capacidade máxima de suporte, ou seja, se $P > M$. A equação que +modela essa sofisticação é +% + $$\fr{dP}{dt} = kP\left(1 - \fr{P}{M}\right).$$ +% +Observe como o fator $(1 - P/M)$ funciona como uma correção ao modelo +simplista. Se a população estiver acima da capacidade de suporte, +então $(1 - P/M) < 0$ e assim $dP/dt < 0$, ou seja, a população decai. +Se porventura a população estiver abaixo da capacidade de suporte, +então $(1 - P/M) > 0$ e assim $dP/dt > 0$. + +Isso não significa que o modelo faz a população oscilar ao redor de +$M$. Não ocorre essa oscilação. O que ocorre é que se você começar com +um $P_0$ abaixo de $M$, então a função-solução tende a $M$ por baixo +quando $t$ cresce sem limite. Similarmente, se você começar com uma +população inicial acima de $M$, então a função-solução tende a $M$ por +cima quando $t$ cresce sem limite. Agora, de fato existe uma solução +que é $P(t) = M$ em que a população permanece em $M$ pra sempre, o que +é chamado de {\em solução de equilíbrio}. Outra solução de equilíbrio +para o modelo logístico é $P(t) = 0$, quando a população permanece em +zero pra sempre. + +Naturalmente a gente se pergunta agora qual é a função-solução. O +modelo logístico é uma equação diferencial separável de primeira +ordem, então sua solução não é difícil de ser obtida se você tiver um +mínimo de técnicas de integração. + \newpage \tableofcontents \newpage \printbibliography