From e0e816946ce932d9e25af39b89c6920b8f148ebf Mon Sep 17 00:00:00 2001
From: Scarlett Gauthier <s.s.gauthier@student.tudelft.nl>
Date: Wed, 19 Aug 2020 10:16:02 +0000
Subject: [PATCH] Add up to figure on page 6
---
src/differential_equations_2.md | 16 ++++++++++++++++
1 file changed, 16 insertions(+)
diff --git a/src/differential_equations_2.md b/src/differential_equations_2.md
index ca9401f..4081c79 100644
--- a/src/differential_equations_2.md
+++ b/src/differential_equations_2.md
@@ -334,6 +334,22 @@ To summarize, this process has broken one partial differential equation into two
ordinary differential equations of different variables. In order to do this, we
needed to introduce a separation constant, which remains to be determined.
+### Boundary and eigenvalue problems ###
+
+Continuing on with the Schr\"{o}dinger equation example from the previous
+section, let us focus on
+
+$$-\frac{\hbar^2}{2m} \phi''(x) = \lambda \phi(x),$$
+$$\phi(0)=\phi(L)=0.$$
+
+This has the form of an eigenvalue equation, in which $\lambda$ is the
+eigenvalue, $- \frac{\hbar^2}{2m} \frac{d^2}{dx^2}[\cdot]$ is the linear
+operator and $\phi(x)$ is the eigenfunction.
+
+Notice that when stating the ordinary differential equation, it is specified
+along with it's boundary conditions. Note that in contrast to an initial value
+problem, a boundary value problem does not always have a solution.
+
--
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