Math 224-01: Differential Equations: Reading Homework 3.6

1. Driven Constant Coefficient Linear ODEs : what is accomplished in this section?
   
   1. Properties of Polynomial Operators : what is the linearity property? demonstrate this with P(D) = D²+3 and the functions y₁ = e^t and y₂ = sin(2t). what other properties do these operators have?
      
      1. Point : how may the general solution to P(D)[y] = f(t) be written? on what property does this rely?
         
      2. Point : how do we find the general solution to P(D)[y] = f(t)?
         
   2. Method of Undetermined Coefficients : what is the particular solution to (D² - 2D + 1)[y] = 3e^-t? 3e^t? y'' - y' - 2y = 4t?
      
      1. Point : what guesses might we use for a particular solution for (D² + aD + b)[y] = tⁿ?
         
      2. Point : how can we get real-valued solutions to differential equations with cosine or sine driving terms?
         
      3. Point : what happens if f(t) isn't a polynomial-exponential or if the polynomial operator has nonconstant coefficients?