Math 224-01: Differential Equations: Reading Homework 1.7

1. Planar Systems and First-Order ODEs : what is different about the types of differential equations we consider in this section as compared to what we've done so far?
   
   1. Systems : why might we be prompted to consider systems, anyway?
      
      1. Point : what is the state space when considering systems?
         
      2. Point : what is an autonomous system?
         
      3. Point : what is a solution to a system? an orbit? component curves?
         
   2. ODEs as Systems : how is the ODE N(x,y) y'(x) + M(x,y) = 0 rewritten as a system? how is dy/dx related to dy/dt and dx/dt, and how does this allow us to rewrite the ODE as a system?
      
   3. Reduction Methods : what does reduction do to a second order differential equation?
      
      1. Point : what substitution reduces a differential equation with y'', y' and no ys to a first-order system?
         
      2. Point : what substitution reduces a differential equation with y'', y', y and no ts to a first-order system?