DifferentialEquationsLecture2

Add lecture notes from the second lecture on differential equations.

src/differential_equations_2.md

-# Linear $n-th$ orer differential equations
\ No newline at end of file
+# Higher order linear differential equations
+
+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
+facilitate this change we are going to change notation. In the previous lecture
+we wrote differential equations for $x(t)$. In this lecture we will write DE's 
+of $y(x)$, where $y$ is an unknown function and $x$ is the independent variable. 
+For this purpose we make the following definitions,
+
+$$y' = \frac{dy}{dx}, \ y'' = \frac{d^2 y}{dx^2}, \ \cdots, \ y^{(n)} = \frac{d^n y}{dx^n}.$$
+