 4.5.4.1.183: In 110 write the given differential equation in the form L(y) g(x),...
 4.5.4.1.184: In 110 write the given differential equation in the form L(y) g(x),...
 4.5.4.1.185: In 110 write the given differential equation in the form L(y) g(x),...
 4.5.4.1.186: In 110 write the given differential equation in the form L(y) g(x),...
 4.5.4.1.187: In 110 write the given differential equation in the form L(y) g(x),...
 4.5.4.1.188: In 110 write the given differential equation in the form L(y) g(x),...
 4.5.4.1.189: In 110 write the given differential equation in the form L(y) g(x),...
 4.5.4.1.190: In 110 write the given differential equation in the form L(y) g(x),...
 4.5.4.1.191: In 110 write the given differential equation in the form L(y) g(x),...
 4.5.4.1.192: In 110 write the given differential equation in the form L(y) g(x),...
 4.5.4.1.193: In 1114 verify that the given differential operator annihilates the...
 4.5.4.1.194: In 1114 verify that the given differential operator annihilates the...
 4.5.4.1.195: In 1114 verify that the given differential operator annihilates the...
 4.5.4.1.196: In 1114 verify that the given differential operator annihilates the...
 4.5.4.1.197: In 1526 find a linear differential operator that annihilates the gi...
 4.5.4.1.198: In 1526 find a linear differential operator that annihilates the gi...
 4.5.4.1.199: In 1526 find a linear differential operator that annihilates the gi...
 4.5.4.1.200: In 1526 find a linear differential operator that annihilates the gi...
 4.5.4.1.201: In 1526 find a linear differential operator that annihilates the gi...
 4.5.4.1.202: In 1526 find a linear differential operator that annihilates the gi...
 4.5.4.1.203: In 1526 find a linear differential operator that annihilates the gi...
 4.5.4.1.204: In 1526 find a linear differential operator that annihilates the gi...
 4.5.4.1.205: In 1526 find a linear differential operator that annihilates the gi...
 4.5.4.1.206: In 1526 find a linear differential operator that annihilates the gi...
 4.5.4.1.207: In 1526 find a linear differential operator that annihilates the gi...
 4.5.4.1.208: In 1526 find a linear differential operator that annihilates the gi...
 4.5.4.1.209: In 2734 find linearly independent functions that are annihilated by...
 4.5.4.1.210: In 2734 find linearly independent functions that are annihilated by...
 4.5.4.1.211: In 2734 find linearly independent functions that are annihilated by...
 4.5.4.1.212: In 2734 find linearly independent functions that are annihilated by...
 4.5.4.1.213: In 2734 find linearly independent functions that are annihilated by...
 4.5.4.1.214: In 2734 find linearly independent functions that are annihilated by...
 4.5.4.1.215: In 2734 find linearly independent functions that are annihilated by...
 4.5.4.1.216: In 2734 find linearly independent functions that are annihilated by...
 4.5.4.1.217: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.218: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.219: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.220: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.221: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.222: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.223: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.224: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.225: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.226: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.227: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.228: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.229: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.230: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.231: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.232: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.233: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.234: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.235: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.236: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.237: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.238: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.239: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.240: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.241: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.242: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.243: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.244: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.245: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.246: In 3564 solve the given differential equation by undetermined coeff...
 4.5.4.1.247: In 6572 solve the given initialvalue problem.y 64y 16, y(0) 1, y(0) 0
 4.5.4.1.248: In 6572 solve the given initialvalue problem.y y x, y(0) 1, y(0) 0
 4.5.4.1.249: In 6572 solve the given initialvalue problem.y 5y x 2, y(0) 0, y(0) 2
 4.5.4.1.250: In 6572 solve the given initialvalue problem.y 5y 6y 10e2x, y(0) 1...
 4.5.4.1.251: In 6572 solve the given initialvalue problem.y y 8 cos 2x 4 sin x,...
 4.5.4.1.252: In 6572 solve the given initialvalue problem.y 2y y xex 5, y(0) 2,...
 4.5.4.1.253: In 6572 solve the given initialvalue problem.y 4y 8y x3, y(0) 2, y...
 4.5.4.1.254: In 6572 solve the given initialvalue problem.y(4) y x ex, y(0) 0, ...
 4.5.4.1.255: Suppose L is a linear differential operator that factors but has va...
Solutions for Chapter 4.5: HigherOrder Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 4.5: HigherOrder Differential Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Chapter 4.5: HigherOrder Differential Equations includes 73 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 73 problems in chapter 4.5: HigherOrder Differential Equations have been answered, more than 23458 students have viewed full stepbystep solutions from this chapter. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069.

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).

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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).

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.

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