From f9ec75cee30950ae602513e2a3ceef3c8b042d07 Mon Sep 17 00:00:00 2001
From: Scarlett Gauthier <s.s.gauthier@student.tudelft.nl>
Date: Wed, 9 Sep 2020 09:21:36 +0000
Subject: [PATCH] Add remaining exercises.
---
src/differential_equations_2.md | 17 +++++++++++++++++
1 file changed, 17 insertions(+)
diff --git a/src/differential_equations_2.md b/src/differential_equations_2.md
index 65480d3..a2d732c 100644
--- a/src/differential_equations_2.md
+++ b/src/differential_equations_2.md
@@ -557,4 +557,21 @@ necessary to work with numerical methods of solution.
solution (we know from the lecture that the operator $\frac{d^2}{dx^2}$ is
hermitian - you can thus directly assume that the solutions form an orthogonal
basis).
+
+5. [:smirk:] Take the partial differential equation
+
+ $$\frac{\partial h(x,y)}{\partial x} + x \frac{\partial h(x,y)}{\partial y} = 0. $$
+
+ Try to make a separation ansatz $h(x,y)=f(x)g(y)$. What do you observe?
+
+6. [:sweat:] *Bonus question - this kind of question will not be asked in the exam*
+
+ We consider the Hilbert space of functions $f(x)$ defined for $x \ \epsilon \ [0,L]$
+ with $f(0)=f(L)=0$.
+
+ Which of the following operators on this space is hermitian?
+
+ (a) Lf = A(x) \frac{d^2 f}{dx^2}
+
+ (b) Lf = \frac{d}{dx} \big{()} A(x) \frac{df}{dx} \big{)}
\ No newline at end of file
--
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