diff --git a/documents/by-course/math-4b/course-notes/dvd.typ b/documents/by-course/math-4b/course-notes/dvd.typ
deleted file mode 100644
index 06deb27..0000000
--- a/documents/by-course/math-4b/course-notes/dvd.typ
+++ /dev/null
@@ -1,341 +0,0 @@
-#import "@preview/ctheorems:1.1.3": *
-#import "@preview/showybox:2.0.3": showybox
-
-#let colors = (
-  rgb("#9E9E9E"),
-  rgb("#F44336"),
-  rgb("#E91E63"),
-  rgb("#9C27B0"),
-  rgb("#673AB7"),
-  rgb("#3F51B5"),
-  rgb("#2196F3"),
-  rgb("#03A9F4"),
-  rgb("#00BCD4"),
-  rgb("#009688"),
-  rgb("#4CAF50"),
-  rgb("#8BC34A"),
-  rgb("#CDDC39"),
-  rgb("#FFEB3B"),
-  rgb("#FFC107"),
-  rgb("#FF9800"),
-  rgb("#FF5722"),
-  rgb("#795548"),
-  rgb("#9E9E9E"),
-)
-
-#let dvdtyp(
-  title: "",
-  subtitle: "",
-  author: "",
-  abstract: none,
-  bibliography: none,
-  paper-size: "a4",
-  date: "today",
-  body,
-) = {
-  set document(title: title, author: author)
-
-  set std.bibliography(style: "springer-mathphys", title: [References])
-
-  show: thmrules
-
-  set page(
-    numbering: "1",
-    number-align: center,
-    header: locate(loc => {
-      if loc.page() == 1 {
-        return
-      }
-      box(stroke: (bottom: 0.7pt), inset: 0.4em)[#text(
-          font: "New Computer Modern",
-        )[
-          *#author* --- #datetime.today().display("[day] [month repr:long] [year]")
-          #h(1fr)
-          *#title*
-        ]]
-    }),
-    paper: paper-size,
-    // The margins depend on the paper size.
-    margin: (
-      left: (86pt / 216mm) * 100%,
-      right: (86pt / 216mm) * 100%,
-    ),
-  )
-
-
-  set heading(numbering: "1.")
-  show heading: it => {
-    set text(font: "Libertinus Serif")
-
-    block[
-      #if it.numbering != none {
-        text(rgb("#2196F3"), weight: 500)[#sym.section]
-
-        text(rgb("#2196F3"))[#counter(heading).display() ]
-      }
-      #it.body
-      #v(0.5em)
-    ]
-  }
-
-  set text(font: "New Computer Modern", lang: "en")
-
-  show math.equation: set text(weight: 400)
-
-
-  // Title row.
-  align(center)[
-    #set text(font: "Libertinus Serif")
-    #block(text(weight: 700, 26pt, title))
-
-
-    #if subtitle != none [#text(12pt, weight: 500)[#(
-          subtitle
-        )]]
-
-    #if author != none [#text(16pt)[#smallcaps(author)]]
-    #v(1.2em, weak: true)
-
-    #if date == "today" {
-      datetime.today().display("[day] [month repr:long] [year]")
-    } else {
-      date
-    }
-
-  ]
-
-  if abstract != none [
-    #v(2.2em)
-    #set text(font: "Libertinus Serif")
-    #pad(x: 14%, abstract)
-    #v(1em)
-  ]
-
-  set outline(fill: repeat[~.], indent: 1em)
-
-  show outline: set heading(numbering: none)
-  show outline: set par(first-line-indent: 0em)
-
-  show outline.entry.where(level: 1): it => {
-    text(font: "Libertinus Serif", rgb("#2196F3"))[#strong[#it]]
-  }
-  show outline.entry: it => {
-    h(1em)
-    text(font: "Libertinus Serif", rgb("#2196F3"))[#it]
-  }
-
-
-  // Main body.
-  set par(
-    justify: true,
-    spacing: 0.65em,
-    first-line-indent: 2em,
-  )
-
-  body
-
-  // Display the bibliography, if any is given.
-  if bibliography != none {
-    show std.bibliography: set text(footnote-size)
-    show std.bibliography: set block(above: 11pt)
-    show std.bibliography: pad.with(x: 0.5pt)
-    bibliography
-  }
-}
-
-#let thmtitle(t, color: rgb("#000000")) = {
-  return text(
-    font: "Libertinus Serif",
-    weight: "semibold",
-    fill: color,
-  )[#t]
-}
-#let thmname(t, color: rgb("#000000")) = {
-  return text(font: "Libertinus Serif", fill: color)[(#t)]
-}
-
-#let thmtext(t, color: rgb("#000000")) = {
-  let a = t.children
-  if (a.at(0) == [ ]) {
-    a.remove(0)
-  }
-  t = a.join()
-
-  return text(font: "New Computer Modern", fill: color)[#t]
-}
-
-#let thmbase(
-  identifier,
-  head,
-  ..blockargs,
-  supplement: auto,
-  padding: (top: 0.5em, bottom: 0.5em),
-  namefmt: x => [(#x)],
-  titlefmt: strong,
-  bodyfmt: x => x,
-  separator: [. \ ],
-  base: "heading",
-  base_level: none,
-) = {
-  if supplement == auto {
-    supplement = head
-  }
-  let boxfmt(name, number, body, title: auto, ..blockargs_individual) = {
-    if not name == none {
-      name = [ #namefmt(name)]
-    } else {
-      name = []
-    }
-    if title == auto {
-      title = head
-    }
-    if not number == none {
-      title += " " + number
-    }
-    title = titlefmt(title)
-    body = [#pad(top: 2pt, bodyfmt(body))]
-    pad(
-      ..padding,
-      showybox(
-        width: 100%,
-        radius: 0.3em,
-        breakable: true,
-        padding: (top: 0em, bottom: 0em),
-        ..blockargs.named(),
-        ..blockargs_individual.named(),
-        [
-          #title#name#titlefmt(separator)#body
-        ],
-      ),
-    )
-  }
-
-  let auxthmenv = thmenv(
-    identifier,
-    base,
-    base_level,
-    boxfmt,
-  ).with(supplement: supplement)
-
-  return auxthmenv.with(numbering: "1.1")
-}
-
-#let styled-thmbase = thmbase.with(
-  titlefmt: thmtitle,
-  namefmt: thmname,
-  bodyfmt: thmtext,
-)
-
-#let builder-thmbox(color: rgb("#000000"), ..builderargs) = styled-thmbase.with(
-  titlefmt: thmtitle.with(color: color.darken(30%)),
-  bodyfmt: thmtext.with(color: color.darken(70%)),
-  namefmt: thmname.with(color: color.darken(30%)),
-  frame: (
-    body-color: color.lighten(92%),
-    border-color: color.darken(10%),
-    thickness: 1.5pt,
-    inset: 1.2em,
-    radius: 0.3em,
-  ),
-  ..builderargs,
-)
-
-#let builder-thmline(
-  color: rgb("#000000"),
-  ..builderargs,
-) = styled-thmbase.with(
-  titlefmt: thmtitle.with(color: color.darken(30%)),
-  bodyfmt: thmtext.with(color: color.darken(70%)),
-  namefmt: thmname.with(color: color.darken(30%)),
-  frame: (
-    body-color: color.lighten(92%),
-    border-color: color.darken(10%),
-    thickness: (left: 2pt),
-    inset: 1.2em,
-    radius: 0em,
-  ),
-  ..builderargs,
-)
-
-#let problem-style = builder-thmbox(
-  color: colors.at(11),
-  shadow: (offset: (x: 2pt, y: 2pt), color: luma(70%)),
-)
-
-#let exercise = problem-style("item", "Exercise")
-#let problem = exercise
-
-#let theorem-style = builder-thmbox(
-  color: colors.at(6),
-  shadow: (offset: (x: 3pt, y: 3pt), color: luma(70%)),
-)
-
-#let example-style = builder-thmbox(
-  color: colors.at(16),
-  shadow: (offset: (x: 3pt, y: 3pt), color: luma(70%)),
-)
-
-#let theorem = theorem-style("item", "Theorem")
-#let lemma = theorem-style("item", "Lemma")
-#let corollary = theorem-style("item", "Corollary")
-
-#let definition-style = builder-thmline(color: colors.at(8))
-
-// #let definition = definition-style("definition", "Definition")
-#let proposition = definition-style("item", "Proposition")
-#let remark = definition-style("item", "Remark")
-#let observation = definition-style("item", "Observation")
-
-// #let example-style = builder-thmline(color: colors.at(16))
-
-#let example = example-style("item", "Example")
-
-#let proof(body, name: none) = {
-  v(0.5em)
-  [_Proof_]
-  if name != none {
-    [ #thmname[#name]]
-  }
-  [.]
-  body
-  h(1fr)
-
-  // Add a word-joiner so that the proof square and the last word before the
-  // 1fr spacing are kept together.
-  sym.wj
-
-  // Add a non-breaking space to ensure a minimum amount of space between the
-  // text and the proof square.
-  sym.space.nobreak
-
-  $square.stroked$
-  v(0.5em)
-}
-
-#let fact = thmplain(
-  "item",
-  "Fact",
-  titlefmt: content => [*#content.*],
-  namefmt: content => [_(#content)._],
-  separator: [],
-  inset: 0pt,
-  padding: (bottom: 0.5em, top: 0.5em),
-)
-#let abuse = thmplain(
-  "item",
-  "Abuse of Notation",
-  titlefmt: content => [*#content.*],
-  namefmt: content => [_(#content)._],
-  separator: [],
-  inset: 0pt,
-  padding: (bottom: 0.5em, top: 0.5em),
-)
-#let definition = thmplain(
-  "item",
-  "Definition",
-  titlefmt: content => [*#content.*],
-  namefmt: content => [_(#content)._],
-  separator: [],
-  inset: 0pt,
-  padding: (bottom: 0.5em, top: 0.5em),
-)
diff --git a/documents/by-course/math-4b/course-notes/main.typ b/documents/by-course/math-4b/course-notes/main.typ
index eabf3a0..6de8a5d 100644
--- a/documents/by-course/math-4b/course-notes/main.typ
+++ b/documents/by-course/math-4b/course-notes/main.typ
@@ -530,3 +530,259 @@ We can find a few particular solutions to our ODE, but how can we find all of th
   Now we have a system and we can solve it using standard linear algebra
   techniques.
 ]
+
+= Principle of superposition, Wronskian complex roots
+
+== Review
+
+Recall:
+
+Second order ODE:
+
+$
+  y'' = F(t,y,y')
+$
+
+Linear:
+
+$
+  y'' + p(t) y' + q(t) y = g(t)
+$
+
+Homogenous:
+
+$
+  y'' + p(t) y' + q(t) y = 0
+$
+
+Constant coefficients: $a y'' + b y' + c y = 0$, with characteristic equation
+
+$
+  a r^2 + b r + c = 0
+$
+
+From this characteristic equation we determine either distinct real roots,
+complex real roots, or repeated real roots. We already know what to do with
+distinct real roots.
+
+$
+  y'' + p(t) y' + q(t) y = 0
+$
+
+The linear combination
+
+$
+  y(t) = c_1 y_1 (t) + c_2 y_2 (t)
+$
+
+is a solution for any constants $c_1, c_2$. The solutions from a vector space!
+Key: equation is linear and homogenous.
+
+#example[
+  Consider the linear, homogenous equation
+
+  $
+    y'' + 5y' + 6y = 0
+  $
+
+  We have two solutions:
+
+  $
+    y_1 (t) e^(-2t) + c_2 e^(-3t)
+  $
+
+  is a general solution. Are these all the solutions?
+]
+
+== The Wronskian
+
+Given any initial values
+
+$
+  y(t_0) y_0, y' (t_0) = y'_0
+$
+
+Substitute in:
+
+$
+  c_1 y_1(t_0) + c_2 y_2(t_0) = y_0 \
+  c_1 y'_1 (t_0) + c_2 y'_2 (t_0) = y'_0 \
+$
+
+We have a linear system for $c_1, c_2$:
+
+$
+  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)
+$
+
+The linear system has a unique solution provided the determinant of the
+coefficient matrix is nonzero:
+
+$
+  mat(y_1(t_0), y_2(t_0); y'_1(t_0), y'_2(t_0)) != 0
+$
+
+$ W = y_1(t_0) y'_2(t_0) - y_2(t_0) y'_1(t_0) != 0 $
+
+#definition[
+  This determinant $W$ is called the *Wronskian* of $y_1(t)$ and $y_2(t)$ at
+  the point $t_0$.
+]
+
+#fact[
+  If $W != 0$ at some point $t_0$, then it is nonzero throughout the interval $I$ where the solution is defined.
+]
+
+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 $cos theta = (c_1)/sqrt(c_1^2 + c_2^2)$