dY dt = 3 1 1 1 Y, Y(0) = 3 1 32.

What are the eigenvalues of the matrix A = 1 0 0 2 ? 2

What are the eigenvalues of the matrix B = 0 1 2 0 ? 3

Compute the general solution of the system dY dt = 3 0 0 2 Y and sketch its phase portrait. 4

Which of the following vectors are eigenvectors for the matrix A = 1 0 2 3 ? (i) Y1 = 0 0 (ii) Y2 = 2 2 (iii) Y3 = 1 1 (iv) Y4 = 0 1 (v) Y5 = 4 4 (vi) Y6 = 1 0 5. Consider t

Consider the harmonic oscillator with mass 1, spring constant 5, and damping coefficient b. Find the values of b for which the system is overdamped, underdamped, critically damped, or undamped.

Compute the equilibrium points for the system dY dt = 0 0 1 1 Y.

Solve the initial-value problem dY dt = 2 37.4 555 8.01234 Y, Y(0) = 0 0 .

For each function y(t), determine if it is a solution to the second-order equation d2 y dt2 + ky = 0, assuming k satisfies the given condition. (a) y(t) = sin kt, k < 0 (b) y(t) = 0, k > 0 (c) y(t) = t 2, k = 0 (d) y(t) = sin k t +2 cos k t, k > 0 (e) y(t) = ekt , k > 0 (f) y(t) = e kt , k < 0

Find a linear system for which the function Y(t) = (3 cos 2t,sin 2t) is a solution.

Compute the general solution of the system dY dt = 0 1 0 0 Y and sketch its phase portrait.

The origin is the only equilibrium point for any linear system.

If Y0 is an eigenvector for a matrix, then so is any nonzero scalar multiple of Y0.

The function Y(t) = (cos 2t,sin t) is not a solution to any linear system.

The graph on the right is the graph of the solution of a damped harmonic oscillator.

The graph on the right is the graph of a typical solution of a damped harmonic oscillator.

If k increases, then the time between successive maxima of solutions of the harmonic oscillator d2 y/dt2 + ky = 0 decreases.

Suppose A is a 22 matrix. The linear system dY/dt = AY can have three different straight-line solutions.

Suppose A is a 22 matrix. No solution of dY/dt = AY can blow up in finite time.

Eight matrices and four phase portraits are given below. For each matrix, form the associated linear system, and determine which system corresponds to each phase portrait. State briefly how you know your choice is correct. You should do this exercise without using technology. (i) 1 1 2 1 (ii) 3 1 1 1 (iii) 3 1 1 0 (iv) 1 1 2 1 (v) 2 0 1 1 (vi) 3 1 1 0 (vii) 0 1 4 4 (viii) 3 3 2 1 (a) 3 3 3 3 x y (b) 3 3 3 3 x y (c) 3 3 3 3 x y (d) 3 3 3 3 x y 20. C

Consider the one-parameter family of linear systems dY dt = 0 3a 1 a Y. (a) Draw the curve in the trace-determinant plane that is obtained from varying the parameter a. (b) Determine all bifurcation values of a and briefly discuss the different types of phase portraits that are exhibited in this one-parameter family.

Eight matrices and four pairs of x(t)- and y(t)-graphs of solutions to linear systems are given below. For each matrix, form the associated linear system, and determine which system corresponds to each pair of graphs. State briefly how you know your choice is correct. You should do this exercise without using technology. (i) 3 4 1 0 (ii) 2 0 1 5 (iii) 3 2 0 1 (iv) 0 1 4 0 (v) 1 2 5 1 (vi) 1 .25 .25 1 (vii) 1.1 2 2 1.1 (viii) 1.1 5 1 0.9 (a) 1234 2 2 t x, y (b) 1234 2 2 t x, y (c) 6 12 18 2 2 t x, y (d) 6 12 18 2 2 t x, y 2

(a) Give an example of a linear system that has an equilibrium solution (x(t), y(t)) such that x(0) = 1 and y(t) = 2x(t) for all t. (b) Give an example of a linear system that has a straight-line solution (x(t), y(t)) such that x(0) = 1 and y(t) = 2x(t) for all t.

d2 y dt2 + 5 dy dt + 6y = 0

d2 y dt2 + 2 dy dt + 5y = 0

d2 y dt2 + 2 dy dt + y = 0 y(0) = 1, y (0) = 1

d2 y dt2 + 2y = 0 y(0) = 3, y (0) = 2

dY dt = 1 3 1 1 Y, Y(0) = 2 3 2

dY dt = 4 2 1 3 Y, Y(0) = 0 1 29.

dY dt = 2 3 2 2 Y, Y(0) = 2 2 30.

dY dt = 3 6 2 1 Y, Y(0) = 7 7 31.

dY dt = 3 1 1 1 Y, Y(0) = 3 1 32.

dY dt = 0 1 2 3 Y, Y(0) = 0 3 Co