Question

1. What is the definition of an eigenvalue and eigenvector of a matrix?

2. Consider the nonhomogeneous equationy′′(t) +y′(t)−6y(t) = 6e2t.

(a)Find the general solution yh(t)of the corresponding homogeneous problem.

(b)Find any particular solution yp(t)of the nonhomogeneous problem using the method of undetermined Coefficients.

c)Find any particular solution yp(t)of the nonhomogeneous problem using the method of variation of Parameters.

(d) What is the general solution of the differential equation?

3. Consider the nonhomogeneous equationy′′(t) + 9y(t) =9cos(3t).

(a)Findt he general solution yh(t)of the corresponding homogeneous problem.(b)Find any particular solution yp(t)of the nonhomogeneous problem.(c) What is the general solution of the differential equation?

4. Determine whether the following statements are TRUE or FALSE.Note: you must write the entire word TRUE or FALSE. You do not need to show your work for this problem.(

a)yp(t) =Acos(t)+Bsin(t)is a suitable guess for the particular solution ofy′′+y= cos(t).(

b)yp(t) =Atetis a suitable guess for the particular solution ofy′′−y=et.

(c)yp(t) =Ae−t2is a suitable guess for the particular solution ofy′′+y=e−t2.(d) The phase portrait of any solution ofy′′+y′+y= 0is a stable spiral.

5. Consider the matrixA=[−2 0 0,0 0 0,0 0−2].(

a) Find theeigenvaluesofA.

(b) Find theeigenvectorsofA.

(c) Does the set of all the eigenvectorsofAform a basis ofR3?

6. Consider the system of differential equationsx′(t) =−2x+y,y′(t) =−5x+ 4y.

a) Write the system in the form~x′=A~x.

b) Find the eigenvalues of.

c) Find theeigenvectorsofA.

d) Find the general solution of this system.

e) Sketch the phase portrait of the system. Label your graphs.

7. Determine whether the following statements are TRUE or FALSE. You must write the entire word “TRUE” or “FALSE’’. You do not need to show your work for this problem.

a) If|A|6= 0 then A does not have a zero eigenvalue.

(b) IfA=[4 2,0 4]then the solution ofx′=Axhas a generalized eigenvector of A.

(c) LetA=[−1 4 0,0 3 3,1 0−2].The sum of the eigenvalues of A is 18.

(d) Let x′=Ax be a 2x2 system. If one of the eigenvalues of A is negative, the stability structure of the equilibrium solution of this system cannot be a stable spiral.

8. Below (next page) are four matrices corresponding to the 2x2 system of equations x′=Ax,where x= (x1, x2). Match each of the four systems (1)–(4) with its corresponding vector field, one of the four plots (A)–(D), on the next page. You do not need to show your work for this problem.

A=[0 1,1−1]

A=[0−1,1 0]

A=[1 2,−2 1]

A=[−1 0,−1−1]

Answer 1