Systems of equations of the form above are a special type of linear system. Linear systems model important applications, such as the spring mass system. Moreover, it is possible to find the general solution for any such linear system. For each of the system of differential equations below, address the following questions:

• How many equilibrium solutions are there are and what are they?

• Are there solutions that, when viewed in the phase plane (i. e., the x − y plane), lie along a straight line? If so, algebraically figure out the exact slope of the straight line(s).

• For those systems that do have solutions that, when viewed in the phase plane, lie along a straight line, figure out the exact x(t) and y(t) equations for any solution with initial condition on the straight line(s).

• For those systems that have straight line solutions, write down the general solution.

• How would you classify the equilibrium solution? Create terms if needed to classify any new types of equilibrium solutions and explain the meaning of your terms.

• For those systems of differential equations that do have solutions that, when viewed in the phase plane, lie along straight lines, what do these straight lines look like in 3D? Provide your best 3D sketch.

(a) dx/dt = −3x + 2y

dy/dt = 6x + y

(b) dx/dt = x + y

dy/dt = −x + y

(c) dx/ dt = −2x − 2y

dy /dt = −x − 3y

(d) dx/ dt = 2x + 2y

dy /dt = x + 3y