diff --git a/src/8_differential_equations_2.md b/src/8_differential_equations_2.md
index 98f09658f7e8c4ebb41655978456ee64469f21db..4408ccc81fbec7ca587a15ff59766e224d6bcb0b 100644
--- a/src/8_differential_equations_2.md
+++ b/src/8_differential_equations_2.md
@@ -10,12 +10,14 @@ The lecture on differential equations consists of three parts, each with their o
 - [Partial differential equations: Separation of variables](#partial-differential-equations-separation-of-variables)
 - [Self-adjoint differential operators](#self-adjoint-differential-operators)
 
-**Total video length:  hour  minutes  seconds**
+**Total video length:  1 hour  9 minutes**
 
 ## Higher order linear differential equations
 
 <iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/ucvIiLgJ2i0?rel=0" frameborder="0" allow="accelerometer; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
 
+### Definitions
+
 In the previous lecture, we focused on first order linear differential equations
 as well as systems of such equations. In this lecture we switch focus to DE's 
 which involve higher derivatives of the function we would like to solve for. To
@@ -40,7 +42,9 @@ $$y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_1 y' + a_0 y = 0. $$
     $$a y_{1}(x) + b y_{2}(x)$$
     
     then linear combination of the solutions is also a solution.
-    
+
+### Mapping to a linear system of first-order DEs
+
 In order to solve a higher order linear DE we will present a trick that makes it
 possible to map the problem of solving a single $n$-th order linear DE into a
 related problem of solving a system of $n$ first order linear DE's. 
@@ -79,8 +83,11 @@ f_1 ' (x) & \cdots & f_{n}'(x) \\
 f^{(n-1)}_{1} (x) & \cdots & f^{(n-1)}_{n} (x) \\
 \end{bmatrix}  \neq 0.$$
 
-The determinant in the preceding line is called the *Wronski determinant*. In 
-particular, to determine solutions, we need to find the eigenvalues of 
+The determinant in the preceding line is called the *Wronski determinant*.
+
+### General solution
+
+To determine particular solutions, we need to find the eigenvalues of 
 
 $$A = \begin{bmatrix} 
 0 & 1 & 0 & \cdots & 0 \\
@@ -408,6 +415,7 @@ obtain the general solution in our example.
 
 <iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/p4MHW0yMMvY?rel=0" frameborder="0" allow="accelerometer; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
 
+### Connection to Hilbert spaces
 
 As we hinted was possible earlier, let us re-write the previous equation by 
 defining a linear operator, $L$, acting on the space of functions which satisfy