DifferentialEquationsLecture2

Add lecture notes from the second lecture on differential equations.

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{)}
     
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