 8.10.1: In 16, find Ly for the given differential operator L and the given ...
 8.10.2: In 16, find Ly for the given differential operator L and the given ...
 8.10.3: In 16, find Ly for the given differential operator L and the given ...
 8.10.4: In 16, find Ly for the given differential operator L and the given ...
 8.10.5: In 16, find Ly for the given differential operator L and the given ...
 8.10.6: In 16, find Ly for the given differential operator L and the given ...
 8.10.7: In 713, determine the general solution to the given differential eq...
 8.10.8: In 713, determine the general solution to the given differential eq...
 8.10.9: In 713, determine the general solution to the given differential eq...
 8.10.10: In 713, determine the general solution to the given differential eq...
 8.10.11: In 713, determine the general solution to the given differential eq...
 8.10.12: In 713, determine the general solution to the given differential eq...
 8.10.13: In 713, determine the general solution to the given differential eq...
 8.10.14: In 1417, determine the annihilator of the given function.F(x) = 5ex...
 8.10.15: In 1417, determine the annihilator of the given function.
 8.10.16: In 1417, determine the annihilator of the given function.F(x) = 2x5...
 8.10.17: In 1417, determine the annihilator of the given function.F(x) = 4x ...
 8.10.18: In 1823, determine a trial solution for the given nonhomogeneous di...
 8.10.19: In 1823, determine a trial solution for the given nonhomogeneous di...
 8.10.20: In 1823, determine a trial solution for the given nonhomogeneous di...
 8.10.21: In 1823, determine a trial solution for the given nonhomogeneous di...
 8.10.22: In 1823, determine a trial solution for the given nonhomogeneous di...
 8.10.23: In 1823, determine a trial solution for the given nonhomogeneous di...
 8.10.24: For 2429, solve the given nonhomogeneous differential equation by u...
 8.10.25: For 2429, solve the given nonhomogeneous differential equation by u...
 8.10.26: For 2429, solve the given nonhomogeneous differential equation by u...
 8.10.27: For 2429, solve the given nonhomogeneous differential equation by u...
 8.10.28: For 2429, solve the given nonhomogeneous differential equation by u...
 8.10.29: For 2429, solve the given nonhomogeneous differential equation by u...
 8.10.30: For 3039, state whether the annihilator method can be used to deter...
 8.10.31: For 3039, state whether the annihilator method can be used to deter...
 8.10.32: For 3039, state whether the annihilator method can be used to deter...
 8.10.33: For 3039, state whether the annihilator method can be used to deter...
 8.10.34: For 3039, state whether the annihilator method can be used to deter...
 8.10.35: For 3039, state whether the annihilator method can be used to deter...
 8.10.36: For 3039, state whether the annihilator method can be used to deter...
 8.10.37: For 3039, state whether the annihilator method can be used to deter...
 8.10.38: For 3039, state whether the annihilator method can be used to deter...
 8.10.39: For 3039, state whether the annihilator method can be used to deter...
 8.10.40: For 4045, use the annihilator method to solve the given differentia...
 8.10.41: For 4045, use the annihilator method to solve the given differentia...
 8.10.42: For 4045, use the annihilator method to solve the given differentia...
 8.10.43: For 4045, use the annihilator method to solve the given differentia...
 8.10.44: For 4045, use the annihilator method to solve the given differentia...
 8.10.45: For 4045, use the annihilator method to solve the given differentia...
 8.10.46: Solve the initialvalue problem: y y 2y = 15e2x , y(0) = 0, y (0) =...
 8.10.47: For 4751, use the variationofparameters method to solve the given...
 8.10.48: For 4751, use the variationofparameters method to solve the given...
 8.10.49: For 4751, use the variationofparameters method to solve the given...
 8.10.50: For 4751, use the variationofparameters method to solve the given...
 8.10.51: For 4751, use the variationofparameters method to solve the given...
 8.10.52: Solve by the reduction of order method, given that y1(x) = emx is a...
 8.10.53: For 5358, find the general solution to the given differential equat...
 8.10.54: For 5358, find the general solution to the given differential equat...
 8.10.55: For 5358, find the general solution to the given differential equat...
 8.10.56: For 5358, find the general solution to the given differential equat...
 8.10.57: For 5358, find the general solution to the given differential equat...
 8.10.58: For 5358, find the general solution to the given differential equat...
 8.10.59: For 5961, solve the given differential equation on the interval x >...
 8.10.60: For 5961, solve the given differential equation on the interval x >...
 8.10.61: For 5961, solve the given differential equation on the interval x >...
 8.10.62: For 6266, determine a particular solution to the given differential...
 8.10.63: For 6266, determine a particular solution to the given differential...
 8.10.64: For 6266, determine a particular solution to the given differential...
 8.10.65: For 6266, determine a particular solution to the given differential...
 8.10.66: For 6266, determine a particular solution to the given differential...
Solutions for Chapter 8.10: Linear Differential Equations of Order n
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 8.10: Linear Differential Equations of Order n
Get Full SolutionsChapter 8.10: Linear Differential Equations of Order n includes 66 full stepbystep solutions. Differential Equations was written by and is associated to the ISBN: 9780321964670. This textbook survival guide was created for the textbook: Differential Equations, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Since 66 problems in chapter 8.10: Linear Differential Equations of Order n have been answered, more than 20281 students have viewed full stepbystep solutions from this chapter.

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).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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 ().

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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