Math 224-01: Differential Equations: Reading Homework 5.1--3

1. First-Order Systems : what is a first-order system of differential equations?
   
   1. From Single ODEs : how do we convert a higher-order ODE into a system?
      
   2. Theory : what is the normal form for a system of n first-order differential equations? how many initial conditions are there for the corresponding IVP? what is the state-space for the system? a time-state curve?
      
      1. vectors : how is vector notation used to represent systems? what is the compact form that this leads to for the normal form of the system?
         
      2. linear systems : what is a linear system of first-order differential equations?
         
   3. Applications : what is the chemical law of mass balance?
      
2. Properties of Systems : what are the state variable, rate function, and initial state for a first-order system? why is this chapter focused on properties of systems?
   
   1. the Fundamental Theorem : what is the fundamental theorem for systems? what are the four parts of it? what do they mean?
      
   2. Autonomous Systems : what is an autonomous system? what are two properties of the solutions to autonomous systems?
      
      1. equilibrium solutions : what are equilibrium solutions to a first-order system? how do we find them?
         
      2. direction fields : what is a direction field for a planar system of first-order ODEs? how is the solution related to this?
         
      3. nullclines : what is a nullcline for a system? how many of them are there?
         
3. Models of Interacting Species :
   
   1. Basic Ideas : what is the balance law? how is it related to populations? what is the population law of mass action?
      
   2. Interaction : what are the seven types of interaction that may characterize interacting species? what differential equations do they give rise to?