Fix problem 5, add nonseparable example.

src/8_differential_equations_2.md

@@ -574,11 +574,11 @@ necessary to work with numerical methods of solution.
 	$\frac{d^2}{dx^2}$ is Hermitian - you can thus directly assume that
 	the solutions form an orthogonal basis). 
 
-5.  [:smirk:] Take the partial differential equation
+5.  [:smirk:] Consider the following partial differential equations, and try to make a separation ansatz $h(x,y)=f(x)g(y)$. What do you observe in each case? (Only attempt the separation, do not solve the problem fully)
 
-    $$\frac{\partial h(x,y)}{\partial x} + x \frac{\partial h(x,y)}{\partial y} = 0. $$
+    (a) $$\frac{\partial h(x,y)}{\partial x} + x \frac{\partial h(x,y)}{\partial y} = 0. $$
 
-    Try to make a separation ansatz $h(x,y)=f(x)g(y)$. What do you observe?
+    (b) $$\frac{\partial h(x,y)}{\partial x} + \frac{\partial h(x,y)}{\partial y} + xy\,h(x,y) = 0$$
 
 6.  [:sweat:] We consider the Hilbert space of functions $f(x)$ defined
     for $x \ \epsilon \ [0,L]$ with $f(0)=f(L)=0$.