 7.5.1: In each of 1 through 4: G a. Draw a direction field. b. Find the ge...
 7.5.2: In each of 1 through 4: G a. Draw a direction field. b. Find the ge...
 7.5.3: In each of 1 through 4: G a. Draw a direction field. b. Find the ge...
 7.5.4: In each of 1 through 4: G a. Draw a direction field. b. Find the ge...
 7.5.5: In each of 5 and 6 the coefficient matrix has a zero eigenvalue. As...
 7.5.6: In each of 5 and 6 the coefficient matrix has a zero eigenvalue. As...
 7.5.7: In each of 7 through 9, find the general solution of the given syst...
 7.5.8: In each of 7 through 9, find the general solution of the given syst...
 7.5.9: In each of 7 through 9, find the general solution of the given syst...
 7.5.10: In each of 10 through 12, solve the given initial value problem. De...
 7.5.11: In each of 10 through 12, solve the given initial value problem. De...
 7.5.12: In each of 10 through 12, solve the given initial value problem. De...
 7.5.13: The system tx_ = Ax is analogous to the secondorder Euler equation...
 7.5.14: Referring to 13, solve the given system of equations in each of 14 ...
 7.5.15: Referring to 13, solve the given system of equations in each of 14 ...
 7.5.16: Referring to 13, solve the given system of equations in each of 14 ...
 7.5.17: In each of 17 through 19, the eigenvalues and eigenvectors of a mat...
 7.5.18: In each of 17 through 19, the eigenvalues and eigenvectors of a mat...
 7.5.19: In each of 17 through 19, the eigenvalues and eigenvectors of a mat...
 7.5.20: Consider a 2 2 system x_ = Ax. If we assume that r1 _= r2, the gene...
 7.5.21: Consider the equation ay__ + by_ + cy = 0, (35) where a, b, and c a...
 7.5.22: The twotank system of in Section 7.1 leads to the initial value pr...
 7.5.23: Consider the system x _ = _ 1 1 1 _ x. a. Solve the system for = 1 ...
 7.5.24: a. Find the general solution of equation (37) if R1 = 1 , R2 = 3 5 ...
 7.5.25: Consider the preceding system of differential equations (37). a. Fi...
Solutions for Chapter 7.5: Homogeneous Linear Systems with Constant Coefficients
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 7.5: Homogeneous Linear Systems with Constant Coefficients
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.5: Homogeneous Linear Systems with Constant Coefficients includes 25 full stepbystep solutions. Since 25 problems in chapter 7.5: Homogeneous Linear Systems with Constant Coefficients have been answered, more than 12383 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.