89 lines
1.7 KiB
Text
89 lines
1.7 KiB
Text
#import "./dvd.typ": *
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#show: dvdtyp.with(
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title: "Math 4B Course Notes",
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author: "Youwen Wu",
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date: "Winter 2025",
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subtitle: [Taught by Guofang Wei],
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)
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#outline()
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= Course logistics
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The textbook is Elementary Differential Equations, 11th edition, 2017. Chapters
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1-4, 6, 7, and 9 will be covered.
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Attendance to discussion sections is mandatory.
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= Lecture #datetime(day: 7, month: 1, year: 2025).display()
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== Trivial preliminaries
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#definition[
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An ODE involves an unknown function of a single variable and its derivatives
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up to some fixed order. The order of an ODE is the order of the highest
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derivative that appears.
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]
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#example[First order ODE][
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$ (dif y) / (dif x) = y^2 $
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]
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#example[Second order ODE][
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$
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y'' &= x \
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integral.double y'' dif x &= integral.double x dif x \
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y &= 1 / 6 x^3 + C
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$
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]
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#definition[
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A function $y(x)$ defined on $(a,b)$ is a *solution* of the ODE
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$ y' = F(x,y) $
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if and only if
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$ y'(x) = F(x, y(x)), forall x in (a,b) $
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]
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#problem[
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Check that $y(x) = 20 + 10e^(-x / 2)$ is a solution to the ODE
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$ y' = -1 / 2 y + 10 $
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]
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#definition[
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A first order ODE
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$ y' = F(x,y) $
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is called *linear* if there are functions $A(x)$ and $B(x)$ such that
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$ F(x,y) = A(x) y + B(x) $
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]
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#example("Linear ODEs")[
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- $y' = x$
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- $y' = y$
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- $y' = x^2$
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]
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#example("Nonlinear ODEs")[
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- $y' = y^2$
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]
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#definition[
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*Equilibrium solutions* for the ODE
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$ y' = F(x,y) $
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are solutions $y(x)$ such that $y'(x) = 0$, that is, $y(x)$ is constant.
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]
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#example[
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The equation
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$ y' = y(y +2) $
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has two equilibria
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$
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y(x) &= 0 \
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y(x) &= -2
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$
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]
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#problem[
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What are the equilibria of the equation
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$ y' = y(y - x) $
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]
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