 3.6.1: d2 y dt2 6 dy dt 7y = 0
 3.6.2: d2 y dt2 dy dt 12y = 0
 3.6.3: d2 y dt2 + 6 dy dt + 9y = 0
 3.6.4: d2 y dt2 4 dy dt + 4y = 0
 3.6.5: d2 y dt2 + 8 dy dt + 25y = 0
 3.6.6: d2 y dt2 4 dy dt + 29y = 0
 3.6.7: d2 y dt2 + 2 dy dt 3y = 0
 3.6.8: d2 y dt2 + 4 dy dt 5y = 0
 3.6.9: d2 y dt2 4 dy dt + 13y = 0
 3.6.10: d2 y dt2 + 4 dy dt + 20y = 0
 3.6.11: d2 y dt2 8 dy dt + 16y = 0 y(0) = 3, y (0) = 11
 3.6.12: d2 y dt2 4 dy dt + 4y = 0 y(0) = 1, y (0) = 1
 3.6.13: m = 1, k = 7, b = 8, with initial conditions y(0) = 1, v(0) = 5
 3.6.14: m = 1, k = 8, b = 6, with initial conditions y(0) = 1, v(0) = 0
 3.6.15: m = 1, k = 5, b = 4, with initial conditions y(0) = 1, v(0) = 0
 3.6.16: m = 1, k = 8, b = 0, with initial conditions y(0) = 1, v(0) = 4
 3.6.17: m = 2, k = 1, b = 3, with initial conditions y(0) = 0, v(0) = 3
 3.6.18: m = 9, k = 1, b = 6, with initial conditions y(0) = 1, v(0) = 1
 3.6.19: m = 2, k = 3, b = 0, with initial conditions y(0) = 2, v(0) = 3
 3.6.20: m = 2, k = 3, b = 1, with initial conditions y(0) = 0, v(0) = 3
 3.6.21: m = 1, k = 7, b = 8, with initial conditions y(0) = 1, v(0) = 5
 3.6.22: m = 1, k = 8, b = 6, with initial conditions y(0) = 1, v(0) = 0
 3.6.23: m = 1, k = 5, b = 4, with initial conditions y(0) = 1, v(0) = 0
 3.6.24: m = 1, k = 8, b = 0, with initial conditions y(0) = 1, v(0) = 4
 3.6.25: m = 2, k = 1, b = 3, with initial conditions y(0) = 0, v(0) = 3
 3.6.26: m = 9, k = 1, b = 6, with initial conditions y(0) = 1, v(0) = 1
 3.6.27: m = 2, k = 3, b = 0, with initial conditions y(0) = 2, v(0) = 3
 3.6.28: m = 2, k = 3, b = 1, with initial conditions y(0) = 0, v(0) = 3
 3.6.29: Construct a table of all the possible harmonic oscillator systems a...
 3.6.30: Suppose y1(t) and y2(t) are solutions of d2 y dt2 + p dy dt + qy = ...
 3.6.31: Suppose y(t) is a complexvalued solution of d2 y dt2 + p dy dt + q...
 3.6.32: Suppose is an eigenvalue for the secondorder equation d2 y dt2 + p...
 3.6.33: Suppose the secondorder equation d2 y dt2 + p dy dt + qy = 0 has 0...
 3.6.34: Consider a harmonic oscillator with mass m = 1 and spring constant ...
 3.6.35: Consider a harmonic oscillator with mass m = 1, spring constant k =...
 3.6.36: An automobiles suspension system consists essentially of large spri...
 3.6.37: Suppose material scientists discover a new type of fluid called mag...
 3.6.38: Consider a harmonic oscillator with m = 1, k = 2, and b = 1. (a) Wh...
 3.6.39: Suppose we wish to make a clock using a mass and a spring sliding o...
 3.6.40: As pointed out in the text, an undamped or underdamped harmonic osc...
Solutions for Chapter 3.6: SECONDORDER LINEAR EQUATIONS
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter 3.6: SECONDORDER LINEAR EQUATIONS
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations 00, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.6: SECONDORDER LINEAR EQUATIONS includes 40 full stepbystep solutions. Since 40 problems in chapter 3.6: SECONDORDER LINEAR EQUATIONS have been answered, more than 16069 students have viewed full stepbystep solutions from this chapter. Differential Equations 00 was written by and is associated to the ISBN: 9780495561989.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Column space C (A) =
space of all combinations of the columns of A.

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.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).