Second-Order Linear Homogeneous
Differential Equations with
Constant Coefficients

Part 2: The Equation y" + by = 0 with b < 0

1. Set b = -1, y(0) = 1, and y'(0) = 0. Plot the graph of the solution of this initial value problem. Repeat for b = -2, -4, and -8. (You may need to increase the range of y for some of the graphs.) How do the solutions change as b varies through this range? Use the symbolic solution to explain what you see in these solution graphs.
2. Reset b to -1. Set the initial conditions to y(0) = 1 and y'(0) = -1, and plot the graph of the solution. Keep b and y(0) the same, but change y'(0), first to -0.9 and then to -1.1, plotting each of the corresponding solutions. Describe how the solutions vary as y'(0) assumes values near -1. Use the symbolic solution to explain what you see in these solution graphs.