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src/differential_equations_1.md

@@ -28,6 +28,7 @@ equation, $n-1$ initial conditions are necessary. To understand why we need
 initial conditions, look at the following example.
 
 !!! check "Example: Initial conditions"
+
     Consider the following calculus problem,
     
     $$\dot{f}(x)=x. $$ 
@@ -48,6 +49,7 @@ Essentially initial conditions are needed when solving differential equations so
 that unknowns resulting from integration may be determined.
 
 !!! info Terminology for Differential Equations
+
     1. If a differential equation does not explicitly contain the 
         independent variable $t$, it is called an *autonomous equation*.
     2. If the largest derivative in a differential equation is of first order, 
@@ -91,6 +93,7 @@ to this type of equation are
 $$x(t) = F(t) + c. $$
 
 !!! check "Example: First order linear differential equation with constant coefficients"
+
     Given the equation
     
     $$\dot{x}(t)=t, $$
@@ -115,6 +118,7 @@ solution! Having a specific form for the function $f(x)$ can often makes it
 possible to solve either implicitly or explicity for the function $x(t)$.
 
 !!! check "Example: Autonomous first order linear differential equation with constant coefficients"
+
     Given the equation
     
     $$\dot{x} = \lambda x, $$
@@ -163,6 +167,7 @@ Given this form of general solution, knowledge of specific functions $f, g$ woul
 make it possible to solve for $x(t)$. 
 
 !!! check "Example: First order linear differential equation with coefficient t"
+
     Let us apply the above strategy to the following equation,
     
     $$\dot{x}= t x^2 .$$
@@ -254,7 +259,7 @@ each arrow. The position of each of the little cars is determined by an initial
 condition. Since the field lines do not cross, and the cars begin on different 
 field lines, they will remain on different field lines. 
 
-![image](figures/Phase_portrait_with_cars.png)
+<img src="figures/Phase_portrait_with_cars.png" width="700">
 
 If $**f**(**x**)$ is not crazy, for axample if it is continuous and 
 differentiable, then it is possible to prove the following two properties for 
@@ -265,7 +270,49 @@ a system of first order linear DE's
     initial condition and the equation i.e. we know where each point "came from"
     $**x**(t'<t)$. 
 
+# Systems of linear first order differential equations
+
+## Homogeneous systems ##
+Any homogeneous system of first order linear DE's can be written in the form 
+
+$$\dot{**x**} = **A**(t) **x**, $$
+
+where $**A**$ is a linear operator. The system is called homogeneous because it 
+does not contain an additional term which is not dependent on $**x**$ (for 
+example an additive constant or an additional function depending only on t). 
+
+An important property of such a system is *linearity*, which has the following 
+implictions
+
+1. If $**x**(t)$ is a solution then $c **x**(t)$ is as well, for some constant c
+2. If $**x**(t)$ and $**y**(t)$ are both solutions, then so is $a **x**(t)+ b **y**(t)$,
+    where a and b are both constants. 
+
+These properties have special importance for modelling physical systems, due to
+the principle of superposition, which is especially important in quantum physics,
+as well as electromagnetism and fluid dynamics. For example in electromagnetism 
+when there are four charges arranged in a square acting on a test charge 
+located within the square, it is sufficient to sum the individual forces in 
+order to find the total force. Physically, this is the principle of superposition, 
+and mathematically superposition is linearity and applies to linear models. 
+
+### General Solution ###
+
+For a system of $n$ linear first order DE's with $n \times n$ linear operator 
+$**A**(t)$, the general solution can be written as 
+
+$$**x**(t) = c_1 **\phi**_1 (t) + c_2 **\phi**_2 (t) + \cdots + c_n **\phi**_n (t),$$
+
+where $\{**\phi**_1 (t), **\phi**_2(t), \cdots, **\phi**_n (t) \}$ are $n$ 
+independent soutions which form a basis for the solution space, and 
+$c_1, c_2, \cdots c_n$ are constants. 
+
+$\{**\phi**_1 (t), **\phi**_2(t), \cdots, **\phi**_n (t) \}$ are a basis if and 
+only if they are linearly independent for fixed $t$:
+
+$$det \big{(}**\phi**_1 (t) | **\phi**_2 (t) | \cdots | **\phi**_n (t) \big{)} \neq 0.$$
 
+If this condition holds for one $t$, it holds for all $t$.