Add phase portrait and existence and uniqueness

src/differential_equations_1.md

@@ -233,6 +233,41 @@ we will develop the general theory for linear equations which will allow us to
 tackle such higher order equations. For now, we move on to considering systems 
 of coupled first order linear DE's. 
 
+# Systems of first order differential equations
+
+An intuitive presentation of a system of coupled first order differential 
+equations can be given by a phase portrait. Before demonstrating such a portrait,
+let us introduce notation that is useful for working with systems of DE's. Several
+coupled DE's can be written down consicely as a single vector equation
+
+$$\dot{**x**}=**f**(**x**). $$
+
+In such an equation the vector $\dot{**x**}$ is the rate of change of a vector 
+quantity, for example the velocity which is the rate of change of the position 
+vector. The term $**f**(**x**)$ describes a vector field, which has one vecter 
+per point $**x**$. This type of equation can also be extended to include a time 
+varying vector field, $**f**(**x**,t)$. 
+
+In the phase portrait below the velocity of the little cars are determined by 
+the vector field $**f**(**x**)$, where the velocity corresponds to the slope of 
+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)
+
+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 
+a system of first order linear DE's
+
+1. *Existence of solution*: For any specified initial condition, there is a solution
+2. *Uniqueness of solution*: Any point $**x**(t)$ is uniquely determined by the
+    initial condition and the equation i.e. we know where each point "came from"
+    $**x**(t'<t)$. 
+
+
+
+