auto-update(nvim): 2025-02-05 01:15:42
This commit is contained in:
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commit
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2 changed files with 256 additions and 341 deletions
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@ -1,341 +0,0 @@
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#import "@preview/ctheorems:1.1.3": *
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#import "@preview/showybox:2.0.3": showybox
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#let colors = (
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rgb("#9E9E9E"),
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rgb("#F44336"),
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rgb("#E91E63"),
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rgb("#9C27B0"),
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rgb("#673AB7"),
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rgb("#3F51B5"),
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rgb("#2196F3"),
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rgb("#03A9F4"),
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rgb("#00BCD4"),
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rgb("#009688"),
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rgb("#4CAF50"),
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rgb("#8BC34A"),
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rgb("#CDDC39"),
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rgb("#FFEB3B"),
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rgb("#FFC107"),
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rgb("#FF9800"),
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rgb("#FF5722"),
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rgb("#795548"),
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rgb("#9E9E9E"),
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)
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#let dvdtyp(
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title: "",
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subtitle: "",
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author: "",
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abstract: none,
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bibliography: none,
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paper-size: "a4",
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date: "today",
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body,
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) = {
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set document(title: title, author: author)
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set std.bibliography(style: "springer-mathphys", title: [References])
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show: thmrules
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set page(
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numbering: "1",
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number-align: center,
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header: locate(loc => {
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if loc.page() == 1 {
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return
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}
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box(stroke: (bottom: 0.7pt), inset: 0.4em)[#text(
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font: "New Computer Modern",
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)[
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*#author* --- #datetime.today().display("[day] [month repr:long] [year]")
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#h(1fr)
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*#title*
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]]
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}),
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paper: paper-size,
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// The margins depend on the paper size.
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margin: (
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left: (86pt / 216mm) * 100%,
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right: (86pt / 216mm) * 100%,
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),
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)
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set heading(numbering: "1.")
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show heading: it => {
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set text(font: "Libertinus Serif")
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block[
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#if it.numbering != none {
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text(rgb("#2196F3"), weight: 500)[#sym.section]
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text(rgb("#2196F3"))[#counter(heading).display() ]
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}
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#it.body
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#v(0.5em)
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]
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}
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set text(font: "New Computer Modern", lang: "en")
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show math.equation: set text(weight: 400)
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// Title row.
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align(center)[
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#set text(font: "Libertinus Serif")
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#block(text(weight: 700, 26pt, title))
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#if subtitle != none [#text(12pt, weight: 500)[#(
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subtitle
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)]]
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#if author != none [#text(16pt)[#smallcaps(author)]]
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#v(1.2em, weak: true)
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#if date == "today" {
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datetime.today().display("[day] [month repr:long] [year]")
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} else {
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date
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}
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]
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if abstract != none [
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#v(2.2em)
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#set text(font: "Libertinus Serif")
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#pad(x: 14%, abstract)
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#v(1em)
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]
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set outline(fill: repeat[~.], indent: 1em)
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show outline: set heading(numbering: none)
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show outline: set par(first-line-indent: 0em)
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show outline.entry.where(level: 1): it => {
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text(font: "Libertinus Serif", rgb("#2196F3"))[#strong[#it]]
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}
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show outline.entry: it => {
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h(1em)
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text(font: "Libertinus Serif", rgb("#2196F3"))[#it]
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}
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// Main body.
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set par(
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justify: true,
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spacing: 0.65em,
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first-line-indent: 2em,
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)
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body
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// Display the bibliography, if any is given.
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if bibliography != none {
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show std.bibliography: set text(footnote-size)
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show std.bibliography: set block(above: 11pt)
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show std.bibliography: pad.with(x: 0.5pt)
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bibliography
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}
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}
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#let thmtitle(t, color: rgb("#000000")) = {
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return text(
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font: "Libertinus Serif",
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weight: "semibold",
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fill: color,
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)[#t]
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}
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#let thmname(t, color: rgb("#000000")) = {
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return text(font: "Libertinus Serif", fill: color)[(#t)]
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}
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#let thmtext(t, color: rgb("#000000")) = {
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let a = t.children
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if (a.at(0) == [ ]) {
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a.remove(0)
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}
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t = a.join()
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return text(font: "New Computer Modern", fill: color)[#t]
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}
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#let thmbase(
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identifier,
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head,
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..blockargs,
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supplement: auto,
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padding: (top: 0.5em, bottom: 0.5em),
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namefmt: x => [(#x)],
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titlefmt: strong,
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bodyfmt: x => x,
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separator: [. \ ],
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base: "heading",
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base_level: none,
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) = {
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if supplement == auto {
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supplement = head
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}
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let boxfmt(name, number, body, title: auto, ..blockargs_individual) = {
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if not name == none {
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name = [ #namefmt(name)]
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} else {
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name = []
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}
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if title == auto {
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title = head
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}
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if not number == none {
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title += " " + number
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}
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title = titlefmt(title)
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body = [#pad(top: 2pt, bodyfmt(body))]
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pad(
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..padding,
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showybox(
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width: 100%,
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radius: 0.3em,
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breakable: true,
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padding: (top: 0em, bottom: 0em),
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..blockargs.named(),
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..blockargs_individual.named(),
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[
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#title#name#titlefmt(separator)#body
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],
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),
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)
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}
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let auxthmenv = thmenv(
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identifier,
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base,
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base_level,
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boxfmt,
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).with(supplement: supplement)
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return auxthmenv.with(numbering: "1.1")
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}
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#let styled-thmbase = thmbase.with(
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titlefmt: thmtitle,
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namefmt: thmname,
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bodyfmt: thmtext,
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)
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#let builder-thmbox(color: rgb("#000000"), ..builderargs) = styled-thmbase.with(
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titlefmt: thmtitle.with(color: color.darken(30%)),
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bodyfmt: thmtext.with(color: color.darken(70%)),
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namefmt: thmname.with(color: color.darken(30%)),
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frame: (
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body-color: color.lighten(92%),
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border-color: color.darken(10%),
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thickness: 1.5pt,
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inset: 1.2em,
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radius: 0.3em,
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),
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..builderargs,
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)
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#let builder-thmline(
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color: rgb("#000000"),
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..builderargs,
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) = styled-thmbase.with(
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titlefmt: thmtitle.with(color: color.darken(30%)),
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bodyfmt: thmtext.with(color: color.darken(70%)),
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namefmt: thmname.with(color: color.darken(30%)),
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frame: (
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body-color: color.lighten(92%),
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border-color: color.darken(10%),
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thickness: (left: 2pt),
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inset: 1.2em,
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radius: 0em,
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),
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..builderargs,
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)
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#let problem-style = builder-thmbox(
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color: colors.at(11),
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shadow: (offset: (x: 2pt, y: 2pt), color: luma(70%)),
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)
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#let exercise = problem-style("item", "Exercise")
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#let problem = exercise
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#let theorem-style = builder-thmbox(
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color: colors.at(6),
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shadow: (offset: (x: 3pt, y: 3pt), color: luma(70%)),
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)
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#let example-style = builder-thmbox(
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color: colors.at(16),
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shadow: (offset: (x: 3pt, y: 3pt), color: luma(70%)),
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)
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#let theorem = theorem-style("item", "Theorem")
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#let lemma = theorem-style("item", "Lemma")
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#let corollary = theorem-style("item", "Corollary")
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#let definition-style = builder-thmline(color: colors.at(8))
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// #let definition = definition-style("definition", "Definition")
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#let proposition = definition-style("item", "Proposition")
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#let remark = definition-style("item", "Remark")
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#let observation = definition-style("item", "Observation")
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// #let example-style = builder-thmline(color: colors.at(16))
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#let example = example-style("item", "Example")
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#let proof(body, name: none) = {
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v(0.5em)
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[_Proof_]
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if name != none {
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[ #thmname[#name]]
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}
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[.]
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body
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h(1fr)
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// Add a word-joiner so that the proof square and the last word before the
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// 1fr spacing are kept together.
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sym.wj
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// Add a non-breaking space to ensure a minimum amount of space between the
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// text and the proof square.
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sym.space.nobreak
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$square.stroked$
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v(0.5em)
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}
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#let fact = thmplain(
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"item",
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"Fact",
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titlefmt: content => [*#content.*],
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||||
namefmt: content => [_(#content)._],
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||||
separator: [],
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||||
inset: 0pt,
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||||
padding: (bottom: 0.5em, top: 0.5em),
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)
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#let abuse = thmplain(
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"item",
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"Abuse of Notation",
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titlefmt: content => [*#content.*],
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||||
namefmt: content => [_(#content)._],
|
||||
separator: [],
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||||
inset: 0pt,
|
||||
padding: (bottom: 0.5em, top: 0.5em),
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||||
)
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#let definition = thmplain(
|
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"item",
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||||
"Definition",
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||||
titlefmt: content => [*#content.*],
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||||
namefmt: content => [_(#content)._],
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||||
separator: [],
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||||
inset: 0pt,
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||||
padding: (bottom: 0.5em, top: 0.5em),
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)
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@ -530,3 +530,259 @@ We can find a few particular solutions to our ODE, but how can we find all of th
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Now we have a system and we can solve it using standard linear algebra
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techniques.
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]
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= Principle of superposition, Wronskian complex roots
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== Review
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Recall:
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Second order ODE:
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$
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y'' = F(t,y,y')
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$
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Linear:
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$
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y'' + p(t) y' + q(t) y = g(t)
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$
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Homogenous:
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$
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y'' + p(t) y' + q(t) y = 0
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$
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Constant coefficients: $a y'' + b y' + c y = 0$, with characteristic equation
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$
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a r^2 + b r + c = 0
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$
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From this characteristic equation we determine either distinct real roots,
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complex real roots, or repeated real roots. We already know what to do with
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distinct real roots.
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$
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y'' + p(t) y' + q(t) y = 0
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$
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The linear combination
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$
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y(t) = c_1 y_1 (t) + c_2 y_2 (t)
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$
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is a solution for any constants $c_1, c_2$. The solutions from a vector space!
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Key: equation is linear and homogenous.
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#example[
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Consider the linear, homogenous equation
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$
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y'' + 5y' + 6y = 0
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$
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We have two solutions:
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$
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y_1 (t) e^(-2t) + c_2 e^(-3t)
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$
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is a general solution. Are these all the solutions?
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]
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== The Wronskian
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Given any initial values
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$
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y(t_0) y_0, y' (t_0) = y'_0
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$
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Substitute in:
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$
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c_1 y_1(t_0) + c_2 y_2(t_0) = y_0 \
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c_1 y'_1 (t_0) + c_2 y'_2 (t_0) = y'_0 \
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$
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We have a linear system for $c_1, c_2$:
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$
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mat(c_1 y_1(t_0), c_2 y_2(t_0); c_1 y'_1 (t_0), c_2 y'_2 (t_0)) vec(c_1, c_2) = vec(y_0, y'_0)
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$
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The linear system has a unique solution provided the determinant of the
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coefficient matrix is nonzero:
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$
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mat(y_1(t_0), y_2(t_0); y'_1(t_0), y'_2(t_0)) != 0
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$
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||||
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$ W = y_1(t_0) y'_2(t_0) - y_2(t_0) y'_1(t_0) != 0 $
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||||
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||||
#definition[
|
||||
This determinant $W$ is called the *Wronskian* of $y_1(t)$ and $y_2(t)$ at
|
||||
the point $t_0$.
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||||
]
|
||||
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||||
#fact[
|
||||
If $W != 0$ at some point $t_0$, then it is nonzero throughout the interval $I$ where the solution is defined.
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||||
]
|
||||
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||||
To summarize our description of the solution space, if $y_1(t)$, $y_2(t)$ are two solutions of the linear homogenous ODE
|
||||
|
||||
$
|
||||
y'' + p(t) y' + q(t) y = 0
|
||||
$
|
||||
|
||||
such that $W(y_1, y_2)$, then the constants $c_1$, $c_2$ can be uniquely determined so that
|
||||
|
||||
$
|
||||
y(t) = c_1 y_1(t) + c_2 y_2 (t)
|
||||
$
|
||||
|
||||
satisfies any initial condition
|
||||
|
||||
$
|
||||
y(t_0) = y_0, y'(t_0) = y'_0
|
||||
$
|
||||
|
||||
$y(t)$ is the general solution.
|
||||
|
||||
== Solution space with constant coefficients, distinct real roots
|
||||
|
||||
Consider the 2nd order homogenous linear diffeq with constant coefficients.
|
||||
|
||||
$
|
||||
a y'' + b y ' + c y = 0
|
||||
$
|
||||
|
||||
Suppose the characteristic equation
|
||||
|
||||
$
|
||||
a r^2 + b r + c = 0
|
||||
$
|
||||
|
||||
has a pair of *distinct real roots* $r_1$, $r_2$. Then we have a pair of solutions
|
||||
|
||||
$
|
||||
y_1(t) = e^(r_1 t), y_2(t) = e^(r_2 t)
|
||||
$
|
||||
|
||||
Question: is this a fundamental set of solutions?
|
||||
|
||||
Check Wronskian.
|
||||
|
||||
$
|
||||
W = det mat(y_1(t_0), y_2(t_0); y'_1(t_0), y'_2(t_0)) = det mat(e^(r_1 t_0), e^(r_2 t_0); r_1 e^(r_1 t_0), r_2 e^(r_2 t_0)) = e^(r_1 t_0) e^(r_2 t_0) (r_2 - r_1) != 0
|
||||
$
|
||||
|
||||
since $r_1 != r_2$.
|
||||
|
||||
The Wronskian of our two solutions is $!= 0$ so the general solution is indeed
|
||||
|
||||
$
|
||||
y(t) = c_1 e^(r_1 t) + c_2 e^(r_2 t)
|
||||
$
|
||||
|
||||
== 2nd order linear homogenous ODE, complex roots
|
||||
|
||||
As usual consider
|
||||
$ a y'' + b y' + c y = 0 $
|
||||
|
||||
with characteristic equation
|
||||
|
||||
$
|
||||
a r^2 + b r + c = 0
|
||||
$
|
||||
|
||||
If $b^2 - 4a c < 0$, then solutions are complex numbers:
|
||||
|
||||
$
|
||||
r_1 = lambda + i_mu, r_2 = lambda - i_mu
|
||||
$
|
||||
|
||||
with
|
||||
|
||||
$
|
||||
lambda = -b / (2a), mu = sqrt(4 a c - b^2) / (2a) != 0
|
||||
$
|
||||
|
||||
Complex solutions
|
||||
|
||||
$
|
||||
z_1(t) = e^(r_1 t) = e^((lambda + i_mu) t) = e^(lambda t) e^(i_mu t), z_2(t) = e^(r_2 t)
|
||||
$
|
||||
|
||||
What is $e^(i_mu t)$?
|
||||
|
||||
Euler's formula:
|
||||
|
||||
$
|
||||
e^(i theta) = cos theta + i sin theta
|
||||
$
|
||||
|
||||
Using Euler's formula we can write
|
||||
|
||||
$
|
||||
z_1(t) = e^(r_1 t) = e^((lambda + i_mu) t) = e^(lambda t) e^(i_mu t) = e^(lambda t) [cos mu t + i sin mu t] \
|
||||
z_2(t) = e^(r_2 t) = e^((lambda - i_mu) t) = e^(lambda t) e^(-i_mu t) = e^(lambda t) [cos mu t - i sin mu t]
|
||||
$
|
||||
|
||||
Define
|
||||
|
||||
$
|
||||
y_1(t) = 1 / 2 [z_1(t) + z_2(t)] = e^(lambda t) cos mu t, "real part of" z_1(t) \
|
||||
y_2(t) = 1 / (2i) [z_1(t) - z_2(t)] = e^(lambda t) sin mu t, "imaginary part of" z_1(t) \
|
||||
$
|
||||
|
||||
By the superposition principle, they are solutions. Are they a fundamental set
|
||||
of solutions? Are they a basis for the solution space?
|
||||
|
||||
Check the Wronskian:
|
||||
|
||||
We see that it is nonzero, therefore, when $b^2 - 4 a c < 0$, the equation
|
||||
|
||||
$
|
||||
a y'' + b y' + c y = 0
|
||||
$
|
||||
|
||||
has two real solutions
|
||||
|
||||
$
|
||||
y_1(t) = e^(lambda t) cos mu t, y_2(t) = e^(lambda t) sin mu t
|
||||
$
|
||||
|
||||
where
|
||||
|
||||
$
|
||||
lambda = -b / (2a), mu = sqrt(4a c - b^2) / (2a) != 0
|
||||
$
|
||||
|
||||
The Wronskian of these solutions is nonzero, so $y_1$ and $y_2$ are a fundamental set of solutions. The general solution of the equation is
|
||||
|
||||
$
|
||||
y(t) = c_1 e^(lambda t) cos mu t + c_2 e^(lambda t) sin mu t
|
||||
$
|
||||
|
||||
== Amplitude and phase angle
|
||||
|
||||
Given $c_1 cos(omega_0 t) + c_2 sin(omega_0 t)$, express it as a single cosine
|
||||
function so we can graph it. Recall this formula:
|
||||
|
||||
$
|
||||
cos(alpha - beta) = cos(alpha) cos(beta) + sin(alpha) sin(beta)
|
||||
$
|
||||
|
||||
Write
|
||||
|
||||
$
|
||||
c_1 cos(omega_0 t) + c_2 sin(omega_0 t) \
|
||||
= sqrt(c_1 ^2 + c_2 ^2) cos(omega_0 t - theta) = A cos(omega_0 t - theta)
|
||||
$
|
||||
|
||||
where $A$ is the amplitude, $theta$ is the phase angle with $cos theta = (c_1)/sqrt(c_1^2 + c_2^2)$
|
||||
|
|
|
|||
Loading…
Reference in a new issue