Math 224-01: Differential Equations: Reading Homework 3.2--3

1. Second-Order ODEs and their Properties : what are we addressing in this section? what do we mean by state variables in second-order ODEs? what does the number mean?
   
   1. the Fundamental Theorem : what is the the fundamental theorem for second-order ODEs? how is it different than the existence/uniqueness theorem for first-order ODEs?
      
      1. graphs : what are a solution curve, component cures, and orbit (trajectory) in the state-plane for a second-order ODE?
         
      2. curves and orbits : what properties to solution curves and orbits have for non-autonomous second-order ODEs? for autonomous second-order ODEs?
         
2. Undriven Constant Coefficient Linear ODEs, I : what are we doing in this section?
   
   1. Finding solution formulas : what is our lucky guess for this type of ODE? what is a characteristic polynomial? what are the characteristic roots of this?
      
      1. solution formulas : what is the solution to the differential equation? how many terms does it have? how many constants?
         
      2. repeated roots : what happens if the characteristic roots are a repeated root?
         
      3. general solutions : what is the general solution to a second-order ODE?
         
   2. the Differentiation Operator : what is the differentiation operator? how do we introduce it by ``factoring'' a differential equation?
      
   3. Superposition : what are the linearity property, principle of superposition, and linear combinations?
      
      1. operator approach to solutions : what is the ``operator approach'' to solving ODEs?