 3.2.1: dY dt = 3 2 0 2 Y
 3.2.2: dY dt = 4 2 1 3 Y 3
 3.2.3: dx dt dy dt = 5 2 1 4 x y 4.
 3.2.4: dx dt dy dt = 2 1 1 4 x y 5.
 3.2.5: dx dt = x 2 dy dt = x y 2
 3.2.6: dx dt = 5x + 4y dy dt = 9x
 3.2.7: dx dt dy dt = 3 4 1 0 x y 8.
 3.2.8: dx dt dy dt = 2 1 1 1 x y Co
 3.2.9: dx dt = 2x + y dy dt = x + y
 3.2.10: dx dt = x 2y dy dt = x 4y
 3.2.11: Solve the initialvalue problem dx dt = 2x 2y dy dt = 2x + y, where...
 3.2.12: Solve the initialvalue problem dx dt = 3x dy dt = x 2y, where the ...
 3.2.13: Solve the initialvalue problem dY dt = 4 1 2 3 Y, Y(0) = Y0, where...
 3.2.14: Solve the initialvalue problem dY dt = 4 2 1 1 Y, Y(0) = Y0, where...
 3.2.15: Show that a is the only eigenvalue and that every nonzero vector is...
 3.2.16: A matrix of the form A = a b 0 d is called upper triangular. Suppos...
 3.2.17: A matrix of the form B = a b b d is called symmetric. Show that B h...
 3.2.18: Compute the eigenvalues of a matrix of the form C = a b c 0 . Compa...
 3.2.19: Consider the secondorder equation d2 y dt2 + p dy dt + qy = 0, whe...
 3.2.20: For the harmonic oscillator with mass m = 1, spring constant k = 4,...
 3.2.21: d2 y dt2 + 7 dy dt + 10y = 0
 3.2.22: d2 y dt2 + 4 dy dt + y = 0
 3.2.23: d2 y dt2 + 4 dy dt + y = 0
 3.2.24: d2 y dt2 + 6 dy dt + 7y = 0
 3.2.25: Verify that the linear system that models the harmonic oscillator w...
Solutions for Chapter 3.2: STRAIGHTLINE SOLUTIONS
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter 3.2: STRAIGHTLINE SOLUTIONS
Get Full SolutionsSince 25 problems in chapter 3.2: STRAIGHTLINE SOLUTIONS have been answered, more than 16314 students have viewed full stepbystep solutions from this chapter. Chapter 3.2: STRAIGHTLINE SOLUTIONS includes 25 full stepbystep solutions. Differential Equations 00 was written by and is associated to the ISBN: 9780495561989. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations 00, edition: 4.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.