Math 224-01: Differential Equations: Reading Homework 1.2

1. Visualizing Solution Curves : what motivation does the book give for using the differential equation to visualize its solution, rather than using a numerical solver? what is a solution to a differential equation?
   
   1. Solution curves : what knowledge allows us to use a numerical solver to find approximate solution curves?
      
      1. existence and uniqueness of solutions : what conditions guarantee the existence of a unique solution to an initial-value problem? what implication does this have any two solution curves to the IVP?
         
      2. geometry of solution curves : what is the geometry behind saying that y(t) is a solution of an ODE? how is a direction field related to this?
         
   2. Nullclines and equilibria :
      
      1. nullclines : what is a nullcline? how does a solution curve cross a nullcline? is a nullcline a solution curve?
         
      2. equlibrium solutions : what is an equilibrium solution to an ODE? how is one related to a nullcline? how would you find an equilibrium solution to an ODE?
         
   3. Compression and zooming : how is it that solution curves generated by a numerical solver can appear to touch?