Math 224-01: Differential Equations: Reading Homework 3.1

1. Second-Order Differential Equations : what's our canonical example of a second-order differential equation model?
   
   1. Springs : what are the forces acting on our example spring? where is y=0 in our model?
      
      1. Point : what is Hooke's law? what is the constant k in this?
         
      2. Point : what is the resulting equation for the motion of the end of a Hooke's law spring?
         
      3. Point : why is y replaced with z - h when introducing a mass m at the end of the spring?
         
   2. Autonomous Models : what is an autonomous ODE? what is an equilibrium solution for a second-order equation?
      
      1. Point : how can a second-order differential equation be rewritten as a system of first-order differential equations?
         
      2. Point : what are: the state space for a second-order differential equation (written as a system), an orbit in the state space, and a state portrait?
         
   3. Linear Approximations : how is a nonlinear equation linearized to obtain a linear approximation?