# 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 = et 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? 3et? 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?

Math 224-01: Differential Equations: Reading Homework 3.6