1. Here are five second order differential equations. Below are five graphs of y(t). Match the letter of the solution graph with the number of the differential equation for which it is a solution. If no graph matches a given differential equation, write None next to the number of the differential equation.

a. y'' + 0.1y' + 2y = √2 cost

b. y'' + y' + y = cos t

c. y'' + 2y = cos(1.3t)

d. y'' + 2y = cos(√2t)

e. y'' + 2y = cos t   

2. Find the solution of the periodically forced mass-spring system given by

y" + 9y = -6 sin(3t)

that satisfies the initial condition y(0) = 0, y'(0) = 1. Then sketch the graph of this solution. Finally, discuss in a couple of sentences the motion of the mass as time goes on.

3. Consider the competing species system of differential equations

dx/dt = (1/400)x(400 - x- (1/2)y)

dy/dt = (1/400)y(400 - y - bx)

where b > 0 is a parameter and x, y ≥ 0.

1. First list all of the equilibrium points for this system. At which values of b do you find a bifurcation?

2. Then sketch the nullclines for this system for values of b before, at, and after each bifurcation value.

3. Next sketch the phase plane before, at, and after each bifurcation value.

4. Finally, in a brief paragraph, describe what happens to the competing populations in each of the above cases.

4. Consider the nonlinear system of differential equations

dx/dt = x - y

dy/dx = y - x³.

1. First find all equilibrium points for this system and determine the type (saddle, sink, etc.) of each using linearization.

2. Then draw the nullclines for this system and indicate which are the x and y-nullclines. Sketch the direction field in the regions between the nullclines.

3. Finally, sketch the solution curves in the phase plane for this system.

5. Answer only-

1. The following system has an equilibrium points at the origin

dx/dt = y - x(x²+y²)

dy/dt = -x - y(x²+y²).

What happen to all nearby solutions?

2. In a sentences or two, described the eventual behavior of solutions of the periodically forced mass-spring system

y'' + y' + y = 3cos(2t).

3. Find one solution of the differential equation

y'' = tan(34).