#import "./dvd.typ": *

#show: dvdtyp.with(
  title: "Math 4B Course Notes",
  author: "Youwen Wu",
  date: "Winter 2025",
  subtitle: [Taught by Guofang Wei],
)

#outline()

= Course logistics

The textbook is Elementary Differential Equations, 11th edition, 2017. Chapters
1-4, 6, 7, and 9 will be covered.

Attendance to discussion sections is mandatory.

= Lecture #datetime(day: 7, month: 1, year: 2025).display()

== Trivial preliminaries

#definition[
  An ODE involves an unknown function of a single variable and its derivatives
  up to some fixed order. The order of an ODE is the order of the highest
  derivative that appears.
]

#example[First order ODE][
  $ (dif y) / (dif x) = y^2 $
]

#example[Second order ODE][
  $
    y'' &= x \
    integral.double y'' dif x &= integral.double x dif x \
    y &= 1 / 6 x^3 + C
  $
]

#definition[
  A function $y(x)$ defined on $(a,b)$ is a *solution* of the ODE
  $ y' = F(x,y) $
  if and only if
  $ y'(x) = F(x, y(x)), forall x in (a,b) $
]

#problem[
  Check that $y(x) = 20 + 10e^(-x / 2)$ is a solution to the ODE
  $ y' = -1 / 2 y + 10 $
]

#definition[
  A first order ODE
  $ y' = F(x,y) $
  is called *linear* if there are functions $A(x)$ and $B(x)$ such that
  $ F(x,y) = A(x) y + B(x) $
]

#example("Linear ODEs")[
  - $y' = x$
  - $y' = y$
  - $y' = x^2$
]

#example("Nonlinear ODEs")[
  - $y' = y^2$
]

#definition[
  *Equilibrium solutions* for the ODE
  $ y' = F(x,y) $
  are solutions $y(x)$ such that $y'(x) = 0$, that is, $y(x)$ is constant.
]

#example[
  The equation
  $ y' = y(y +2) $
  has two equilibria
  $
    y(x) &= 0 \
    y(x) &= -2
  $
]

#problem[
  What are the equilibria of the equation
  $ y' = y(y - x) $
]