From 228eb4c48ac18b00ea3b517f83c2413aad9060ad Mon Sep 17 00:00:00 2001
From: Michael Wimmer <m.t.wimmer@tudelft.nl>
Date: Tue, 15 Sep 2020 21:46:20 +0200
Subject: [PATCH] add some more subheadings
---
src/8_differential_equations_2.md | 16 ++++++++++++----
1 file changed, 12 insertions(+), 4 deletions(-)
diff --git a/src/8_differential_equations_2.md b/src/8_differential_equations_2.md
index 98f0965..4408ccc 100644
--- a/src/8_differential_equations_2.md
+++ b/src/8_differential_equations_2.md
@@ -10,12 +10,14 @@ The lecture on differential equations consists of three parts, each with their o
- [Partial differential equations: Separation of variables](#partial-differential-equations-separation-of-variables)
- [Self-adjoint differential operators](#self-adjoint-differential-operators)
-**Total video length: hour minutes seconds**
+**Total video length: 1 hour 9 minutes**
## Higher order linear differential equations
<iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/ucvIiLgJ2i0?rel=0" frameborder="0" allow="accelerometer; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+### Definitions
+
In the previous lecture, we focused on first order linear differential equations
as well as systems of such equations. In this lecture we switch focus to DE's
which involve higher derivatives of the function we would like to solve for. To
@@ -40,7 +42,9 @@ $$y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_1 y' + a_0 y = 0. $$
$$a y_{1}(x) + b y_{2}(x)$$
then linear combination of the solutions is also a solution.
-
+
+### Mapping to a linear system of first-order DEs
+
In order to solve a higher order linear DE we will present a trick that makes it
possible to map the problem of solving a single $n$-th order linear DE into a
related problem of solving a system of $n$ first order linear DE's.
@@ -79,8 +83,11 @@ f_1 ' (x) & \cdots & f_{n}'(x) \\
f^{(n-1)}_{1} (x) & \cdots & f^{(n-1)}_{n} (x) \\
\end{bmatrix} \neq 0.$$
-The determinant in the preceding line is called the *Wronski determinant*. In
-particular, to determine solutions, we need to find the eigenvalues of
+The determinant in the preceding line is called the *Wronski determinant*.
+
+### General solution
+
+To determine particular solutions, we need to find the eigenvalues of
$$A = \begin{bmatrix}
0 & 1 & 0 & \cdots & 0 \\
@@ -408,6 +415,7 @@ obtain the general solution in our example.
<iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/p4MHW0yMMvY?rel=0" frameborder="0" allow="accelerometer; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+### Connection to Hilbert spaces
As we hinted was possible earlier, let us re-write the previous equation by
defining a linear operator, $L$, acting on the space of functions which satisfy
--
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