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- Question : 1.1.1 - Find the general solution and draw the phase portrait for the follow- ing linear systems:Hint: Write (d) as a second-order linear differential equation with constant coefficients, solve it by standard methods, and note that 4+ A =constant on the solution curves. In (e), find x2(t) = c2e and then the x1-equation becomes a first order linear differential equation.
- Question : 1.1.2 - Find the general solution and draw the phase portraits for the fol- lowing three-dimensional linear systems:Hint: In (c), show that the solution curves lie on right circular cylin- ders perpendicular to the x1, x2 plane. Identify the stable and unsta- ble subspaces in (a) and (b). The x3-axis is the stable subspace in (c) and the x1, x2 plane is called the center subspace in (c); cf. Section 1.9.
- Question : 1.1.3 - Find the general solution of the linear system it = xi i2 = axe where a is a constant. Sketch the phase portraits for a = -1,a = 0 and a = 1 and notice that the qualitative structure of the phase portrait is the same for all a < 0 as well as for all a > 0, but that it changes at the parameter value a = 0 called a bifurcation value.
- Question : 1.1.4 - Find the general solution of the linear system (1) when A is the n x n diagonal matrix A = diag[Aj, A21... ,An]- What condition on the eigenvalues Al, ... , An will guarantee that limt_,,. x(t) = 0 for all solutions x(t) of (1)?
- Question : 1.1.5 - What is the relationship between the vector fields defined by x=Ax and x=kAx where k is a non-zero constant? (Describe this relationship both for k positive and k negative.)
- Question : 1.1.6 - (a) If u(t) and v(t) are solutions of the linear system (1), prove that for any constants a and b, w(t) = au(t) + bv(t) is a solution. (b) For _ 11 0 A - 0 -2 ' find solutions u(t) and v(t) of k = Ax such that every solution is a linear combination of u(t) and v(t).
- Question : 1.2.1 - Find the eigenvalues and eigenvectors of the matrix A and show that B = P-'AP is a diagonal matrix. Solve the linear system yy = By and then solve is = Ax using the above corollary. And then sketch the phase portraits in both the x plane and y plane.
- Question : 1.2.2 - Find the eigenvalues and eigenvectors for the matrix A, solve the linear system Sc = Ax, determine the stable and unstable subspaces for the linear system, and sketch the phase portrait for
- Question : 1.2.3 - Write the following linear differential equations with constant coeffi- cients in the form of the linear system (1) and solve:
- Question : 1.2.4 - Using the corollary of this section solve the initial value problem x = Ax x(0) = xo (a) with A given by 1(a) above and xo = (1, 2)T (b) with A given in problem 2 above and xo = (1, 2,3)T.
- Question : 1.2.5 - Let the n x n matrix A have real, distinct eigenvalues. Find conditions on the eigenvalues that are necessary and sufficient for limt.,,. x(t) _ 0 where x(t) is any solution of * = Ax.
- Question : 1.2.6 - Let the n x n matrix A have real, distinct eigenvalues. Let 4i(t, xo) be the solution of the initial value problem is = Ax x(0) = x0. Show that for each fixed t E R, lim 0(t, YO) = 0(t, x0). Yo-xo This shows that the solution 0(t, xo) is a continuous function of the initial condition.
- Question : 1.2.7 - Let the 2 x 2 matrix A have real, distinct eigenvalues A and Ez. Suppose that an eigenvector of A is (1, 0)T and an eigenvector of
- Question : 1.3.1 - Compute the operator norm of the linear transformation defined by the following matrices:Hint: In (c) maximize IAxI2 = 26x + 10xjx2 + x2 subject to the constraint xi + z2 = 1 and use the result of Problem 2; or use the fact that IIAII = [Max eigenvalue of ATA]1/2. Follow this same hint for (b).
- Question : 1.3.2 - Show that the operator norm of a linear transformation T on R" satisfies
- Question : 1.3.3 - Use the lemma in this section to show that if T is an invertible linear transformation then IITII > 0 and
- Question : 1.3.4 - If T is a linear transformation on R" with IIT - III < 1, prove that T is invertible and that the series Ek o(I - T)k converges absolutely to T-'. Hint: Use the geometric series.
- Question : 1.3.5 - Compute the exponentials of the following matrices:
- Question : 1.3.6 - (a) For each matrix in Problem 5 find the eigenvalues of eA. (b) Show that if x is an eigenvector of A corresponding to the eigen- value A, then x is also an eigenvector of eA corresponding to the eigenvalue ea. (c) If A = Pdiag[Aj]P-1, use Corollary 1 to show that det eA = etraceA Also, using the results in the last paragraph of this section, show that this formula holds for any 2 x 2 matrix A.
- Question : 1.3.7 - Compute the exponentials of the following matrices:Hint: Write the matrices in (b) and (c) as a diagonal matrix S plus a matrix N. Show that S and N commute and compute es as in part (a) and eN by using the definition.
- Question : 1.3.8 - Find 2 x 2 matrices A and B such that eA+B # eAeB
- Question : 1.3.8 - Let T be a linear operator on R" that leaves a subspace E C R" invariant; i.e., for all x E E, T(x) E E. Show that eT also leaves E invariant.
- Question : 1.4.1 - Use the forms of the matrix eBt computed in Section 1.3 and the theorem in this section to solve the linear system is = Bx for
- Question : 1.4.2 - Solve the following linear system and sketch its phase portraitThe origin is called a stable focus for this system.
- Question : 1.4.3 - Find eAt and solve the linear system x = Ax for Cf. Problem 1 in Problem Set 2.
- Question : 1.4.4 - Given 1 0 0 Compute the 3 x 3 matrix eAt and solve x = Ax. Cf. Problem 2 in Problem Set 2.
- Question : 1.4.5 - Find the solution of the linear system is = Ax where
- Question : 1.4.6 - Let T be a linear transformation on R" that leaves a subspace E c R" invariant (i.e., for all x E E, T(x) E E) and let T(x) = Ax with respect to the standard basis for R". Show that if x(t) is the solution of the initial value problem is = Ax x(0) = xo with xO E E, then x(t) E E for all t E R.
- Question : 1.4.7 - Suppose that the square matrix A has a negative eigenvalue. Show that the linear system x = Ax has at least one nontrivial solution x(t) that satisfies
- Question : 1.4.8 - (Continuity with respect to initial conditions.) Let
- Question : 1.5.1 - Use the theorem in this section to determine if the linear system x = Ax has a saddle, node, focus or center at the origin and determine the stability of each node or focus:Linear Systems in R2
- Question : 1.5.2 - Solve the linear system is = Ax and sketch the phase portrait for
- Question : 1.5.3 - For what values of the parameters a and b does the linear system x = Ax have a sink at the origin?
- Question : 1.5.4 - If det A = 0, then the origin is a degenerate critical point of x = Ax. Determine the solution and the corresponding phase portraits for the linear system withNote that the origin is not an isolated equilibrium point in these cases. The four different phase portraits determined in (a) with A > 0 or A < 0, (b) and (c) above, together with the sources, sinks, centers and saddles discussed in this section, illustrate the eight different types of qualitative behavior that are possible for a linear system.
- Question : 1.5.5 - Write the second-order differential equation s+ax+bx=0 as a system in R2 and determine the nature of the equilibrium point at the origin.
- Question : 1.5.6 - Find the general solution and draw the phase portrait for the linear systemWhat role do the eigenvectors of the matrix A play in determining the phase portrait? Cf. Case II.
- Question : 1.5.7 - Describe the separatrices for the linear system it = x1 + 2x2 i2 = 3x1 + 4x2. Hint: Find the eigenspaces for A.
- Question : 1.5.8 - Determine the functions r(t) = Ix(t)l and 9(t) = tan-lx2(t)/x1(t) for the linear system
- Question : 1.5.9 - Polar Coordinates) Given the linear system i1=ax1-bx2 i2 = bx1 + axe. Differentiate the equations r2 = xi +x2 and 0 = tan-1(x2/xl) with respect to t in order to obtain r - xlil + x212 and 9 = xli2 x211 r r2 for r 96 0. For the linear system given above, show that these equa- tions reduce to r=ar and 9=b. Solve these equations with the initial conditions r(0) = ro and 0(0) _ 9o and show that the phase portraits in Figures 3 and 4 follow im- mediately from your solution. (Polar coordinates are discussed more thoroughly in Section 2.10 of Chapter 2).
- Question : 1.6.1 - Solve the initial value problem (1) with
- Question : 1.6.2 - Solve the initial value problem (1) with 0 -2 0 A= 1 2 0 . 0 0 -2 Determine the stable and unstable subspaces and sketch the phase portrait
- Question : 1.6.3 - Solve the initial value problem (1) with
- Question : 1.6.4 - Solve the initial value problem (1) with
- Question : 1.7.1 - Solve the initial value problem (1) with the matrix
- Question : 1.7.2 - Solve the initial value problem (1) with the matrix
- Question : 1.7.3 - Solve the initial value problem (1) with the matrix
- Question : 1.7.4 - The "Putzer Algorithm" given below is another method for comput- ing eA0 when we have multiple eigenvalues; cf. (WJ, p. 49. eA` = ri (t)I + r2(t)P1 + - - + rn(t)PP-1 where P1=(A-A1I), P. = (A - A1I)...(A - \.I) and rj(t), j = 1,...,n, are the solutions of the first-order linear differential equations and initial conditions ri = A1r1 with r1(0) = 1, r2 = .12r2 + r1 with r2(0) = 0 rtt = I\nrn + rn-1 with rn(0) = 0. Use the Putzer Algorithm to compute eAl for the matrix A given in (a) Example 1 (b) Example 3 (c) Problem 2(c) (d) Problem 3(b).
- Question : 1.8.1 - Find the Jordan canonical forms for the following matrices
- Question : 1.8.2 - Find the Jordan canonical forms for the following matrices
- Question : 1.8.3 - (a) List the five upper Jordan canonical forms for a 4 x 4 matrix A with a real eigenvalue A of multiplicity 4 and give the corre- sponding deficiency indices in each case. (b) What is the form of the solution of the initial value problem (4) in each of these cases?
- Question : 1.8.4 - (a) What are the four upper Jordan canonical forms for a 4 x 4 matrix A having complex eigenvalues? (b) What is the form of the solution of the initial value problem (4) in each of these cases?
- Question : 1.8.5 - (a) List the seven upper Jordan canonical forms for a 5 x 5 ma- trix A with a real eigenvalue A of multiplicity 5 and give the corresponding deficiency indices in each case.(b) What is the form of the solution of the initial value problem (4) in each of these cases?
- Question : 1.8.6 - Find the Jordan canonical forms for the following matricesFind the solution of the initial value problem (4) for each of these matrices.
- Question : 1.8.7 - Suppose that B is an m x m matrix given by equation (2) and that Q = diag[1, e, e2, ... , e'"_ 1 J. Note that B can be written in the form B=AI+N where N is nilpotent of order m and show that for e > 0 Q-1 BQ = JAI + eN. Suppose that B is an m x m matrix given by equation (2) and that Q = diag[1, e, e2, ... , e'"_ 1 J. Note that B can be written in the form B=AI+N where N is nilpotent of order m and show that for e > 0 Q-1 BQ = JAI + eN.
- Question : 1.8.8 - What are the eigenvalues of a nilpotent matrix N?
- Question : 1.8.9 - Show that if all of the eigenvalues of the matrix A have negative real parts, then for all xo E R" slim x(t) = 0 00 where x(t) is the solution of the initial value problem (4).
- Question : 1.8.10 - Suppose that the elementary blocks B in the Jordan form of the ma- trix A, given by (2) or (3), have no ones or I2 blocks off the diagonal. (The matrix A is called semisimple in this case.) Show that if all of the eigenvalues of A have nonpositive real parts, then for each xa E R" there is a positive constant M such that Ix(t)I < M for all t > 0 where x(t) is the solution of the initial value problem (4).
- Question : 1.8.11 - Show by example that if A is not semisimple, then even if all of the eigenvalues of A have nonpositive real parts, there is an xa E R" such that slim Ix(t)I = 00. 00 Hint: Cf. Example 4 in Section 1.7.
- Question : 1.8.12 - For any solution x(t) of the initial value problem (4) with det A 36 0 and xo 36 0 show that exactly one of the following alternatives holds. (a) slim x(t) = 0 and t limo Ix(t)I = 00; 00 -- (b) slim Ix(t) I = oo and t lim o x(t) = 0; Oc -- (c) There are positive constants m and M such that for all t E R m < Ix(t)I < M; (d) (e) (f) lim Ix(t)I = oo; t-.
- Question : 1.9.1 - Find the stable, unstable and center subspaces E8, E
- Question : 1.9.2 - Same as Problem 1 for the matrices
- Question : 1.9.3 - Solve the system 59 0 2 0 is = -2 0 0 X. 2 0 6 Find the stable, unstable and center subspaces E', E" and E
- Question : 1.9.4 - ind the stable, unstable and center subspaces E', E" and E
- Question : 1.9.5 - et A be an n x n nonsingular matrix and let x(t) be the solution of the initial value problem (1) with x(0) = xo. Show that (a) if xo E E' - {0} then slim x(t) = 0 and a lim o Ix(t)l = 00; 00 -- (b) if xo E E" - {0} then slim jx(t) I = oo and t limox(t) = 0; -00 - (c) if xo E Ec - {0} and A is semisimple (cf. Problem 10 in Sec- tion 1.8), then there are positive constants m and M such that for alltER,m
- Question : 1.9.6 - Show that the only invariant lines for the linear system (1) with x E R2 are the lines ax, + bx2 = 0 where v = (-b, a)T is an eigenvector of A.
- Question : 1.10.1 - Just as the method of variation of parameters can be used to solve a nonhomogeneous linear differential equation, it can also be used to solve the nonhomogeneous linear system (1). To see how this method can be used to obtain the solution in the form (3), assume that the solution x(t) of (1) can be written in the form x(t) = 4)(t)c(t) where 4(t) is a fundamental matrix solution of (2). Differentiate this equation for x(t) and substitute it into (1) to obtain c'(t) = t-1(t)b(t). Integrate this equation and use the fact that c(O) = t'1(0)xo to obtain c(t) = c-1(0)xo + t J -1(r)b(r)dr. 0 Finally, substitute the function c(t) into x(t) = I'(t)c(t) to obtain (3).
- Question : 1.10.2 - Use Theorem 1 to solve the nonhomogeneous linear system x - [ o -1] X + (1) with the initial condition X(0) = ().
- Question : 1.10.3 - Show that [ e -2t cos t - sin t fi(t) e-2t sin t cos tJ is a fundamental matrix solution of the nonautonomous linear system is = A(t)x with A(t) _ 2 cost t -1 -sin 2t - 11 - sin 2t -2 sine t ] Find the inverse of 4(t) and use Theorem 1 and Remark 1 to solve the nonhomogenous linear system is = A(t)x + b(t) with A(t) given above and b(t) = (l,e-2t)T. Note that, in general, if A(t) is a periodic matrix of period T, then corresponding to any fundamental matrix fi(t), there exists a periodic matrix P(t) of period 2T and a constant matrix B such that I'(t) = P(t)eat. Cf. [C/L), p. 81. Show that P(t) is a rotation matrix and B = diag[-2, 0] in this problem.

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