DifferentialEquationsLecture2

Add lecture notes from the second lecture on differential equations.

src/8_differential_equations_2.md

@@ -153,29 +153,6 @@ $$f(x) = e^{\lambda_1 x}, \ x e^{\lambda_1 x} , \ \cdots, \ x^{m_{1}-1} e^{\lamb
     
     $$y'' + Ey = 0.$$ 
     
-    Let us reduce this second order equation to a system of two first order 
-    equations. Define
-    
-    $$y_1=y$$
-    $$y_2=y'.$$
-    
-    Writing $**y**= \begin{bmatrix} 
-    y_1 \\
-    y_2 \\ 
-    \end{bmatrix}$, the DE can be written,
-    
-    $$\dot{**y**} = \begin{bmatrix} 
-    y_2 \\
-    -E y_1 \\
-    \end{bmatrix}$$
-    $$\dot{**y**} = \begin{bmatrix} 
-    0 & 1 \\
-    -E & 0 \\
-    \end{bmatrix} \begin{bmatrix} 
-    y_1 \\
-    y_2 \\
-    \end{bmatrix}.$$
-    
     The characteristic polynomial of this equation is 
     
     $$P(\lambda) = \lambda^2 + E.$$