Add first half of exercises.

src/differential_equations_2.md

@@ -516,4 +516,45 @@ possible approach is to try working in a different coordinate system. There are
 a few more analytic techniques available, however in many situations it becomes
 necessary to work with numerical methods of solution. 
 
+# Problems
+
+1.  [:grinning:] Which of the following equations for $y(x)$ is linear?
+
+        (a) y''' - y'' + x cos(x) y' + y - 1 = 0
+
+        (b) y''' + 4 x y' - cos(x) y = 0
+
+        (c) y'' + y y' = 0
+
+        (d) y'' + e^x y' - x y = 0
+
+2.  [:grinning:] Find the general solution to the equation 
+
+        $$y'' - 4 y' + 4 y = 0. $$
+
+        Show explicitly by computing the Wronski determinant that the 
+        basis for the solution space is actually linearly independent. 
+
+3.  [:grinning:] Find the general solution to the equation 
+
+        $$y''' - y'' + y' - y = 0.$$
+
+        Then find the solution to the initial conditions $y''(0) =0$, $y'(0)=1$, $y(0)=0$. 
+
+4.  [:smirk:] Take the Laplace equation in 2D:
+
+        $$\frac{\partial^2 \phi(x,y)}{\partial x^2} + \frac{\partial^2 \phi(x,y)}{\partial y^2} = 0.$$
+
+        (a) Make a separation ansatz $\phi(x,y) = f(x)g(y)$ and write 
+          down the resulting ordinary differential equations.
+
+        (b) Now assume that the boundary conditions $\phi(0,y) = \phi(L,y) =0$ for 
+          all y, i.e.  f(0)=f(L)=0. Find all solutions $f(x)$ and the corresponding 
+          eigenvalues.
+
+        (c) Finally, for each eigenvalue, find the general solution $g(y)$ for this
+          eigenvalue. Combine this with all solutions $f(x)$ to write down the general
+          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). 
     
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