1. What is the definition of an eigenvalue and eigenvector of amatrix?
2. Consider the nonhomogeneous equationy??(t) +y?(t)?6y(t) =6e2t.
(a)Find the general solution yh(t)of the correspondinghomogeneous problem.
(b)Find any particular solution yp(t)of the nonhomogeneousproblem using the method of undetermined Coefficients.
c)Find any particular solution yp(t)of the nonhomogeneousproblem using the method of variation of Parameters.
(d) What is the general solution of the differentialequation?
3. Consider the nonhomogeneous equationy??(t) + 9y(t)=9cos(3t).
(a)Findt he general solution yh(t)of the correspondinghomogeneous problem.(b)Find any particular solution yp(t)of thenonhomogeneous problem.(c) What is the general solution of thedifferential equation?
4. Determine whether the following statements are TRUE orFALSE.Note: you must write the entire word TRUE or FALSE. You donot need to show your work for this problem.(
a)yp(t) =Acos(t)+Bsin(t)is a suitable guess for the particularsolution ofy??+y= cos(t).(
b)yp(t) =Atetis a suitable guess for the particular solutionofy???y=et.
(c)yp(t) =Ae?t2is a suitable guess for the particular solutionofy??+y=e?t2.(d) The phase portrait of any solution ofy??+y?+y= 0isa 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 basisofR3?
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 yourgraphs.
7. Determine whether the following statements are TRUE or FALSE.You must write the entire word “TRUE” or “FALSE’’. You do not needto 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 generalizedeigenvector of A.
(c) LetA=[?1 4 0,0 3 3,1 0?2].The sum of the eigenvalues of A is18.
(d) Let x?=Ax be a 2x2 system. If one of the eigenvalues of A isnegative, the stability structure of the equilibrium solution ofthis system cannot be a stable spiral.
8. Below (next page) are four matrices corresponding to the 2x2system of equations x?=Ax,where x= (x1, x2). Match each of the foursystems (1)–(4) with its corresponding vector field, one of thefour plots (A)–(D), on the next page. You do not need to show yourwork for this problem.
A=[0 1,1?1]
A=[0?1,1 0]
A=[1 2,?2 1]
A=[?1 0,?1?1]
