Math 224-01: Differential Equations: Reading Homework 3.4

1. Undriven Constant Coefficient Linear ODEs, II : is it odd that we haven't specified ``second-order ODEs'' here? what equations do we obtain solutions for in this section?
   
   1. Complex-Valued Functions and Solutions : what questions do we think about here? what are their answers? how is D[y] defined for complex-valued functions?
      
      1. Point : how is linearity important when considering polynomial operators?
         
      2. Point : how is the exponential function e^{(alpha + ibeta)t} written using Euler's formula?
         
      3. Point : what's D[e^{alpha + ibeta t}]? what does this mean about P(D)[e^rt] for complex values of r?
         
   2. Finding Solution Formulas : how do we conclude that u(t) and v(t), the real and imaginary parts of a solution e^{(alpha+i beta)t} to P(D)[y] = 0, are themselves solutions to P(D)[y] = 0?
      
      1. Point : what are the solutions to P(D)[y] = y'' + a y' + b y = 0 for any a and b?