 2.15.1: Find the general solution of each of the following equations.Find t...
 2.15.2: Find the general solution of each of the following equations.yo'6y...
 2.15.3: Find the general solution of each of the following equations.y@"5y...
 2.15.4: Find the general solution of each of the following equations. y"'y...
 2.15.5: Solve each of the following initialvalue problems.
 2.15.6: Solve each of the following initialvalue problems.
 2.15.7: Solve each of the following initialvalue problems.
 2.15.8: Given that yl(t) = e ' cos t is a solution of ~(~~)2y~~+~"+2~'2y ...
 2.15.9: Find a particular solution of each of the following equations. y"' ...
 2.15.10: Find a particular solution of each of the following equations.
 2.15.11: Find a particular solution of each of the following equations.y"v'+...
 2.15.12: Find a particular solution of each of the following equations.
 2.15.13: Find a particular solution of each of the following equations.
 2.15.14: Find a particular solution of each of the following equations.
 2.15.15: Find a particular solution of each of the following equations.
 2.15.16: Find a particular solution of each of the following equations.
 2.15.17: Find a particular solution of each of the following equations.
 2.15.18: Find a particular solution of each of the following equations.
Solutions for Chapter 2.15: Higherorder equations
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 2.15: Higherorder equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3. Differential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069. Chapter 2.15: Higherorder equations includes 18 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 18 problems in chapter 2.15: Higherorder equations have been answered, more than 5865 students have viewed full stepbystep solutions from this chapter.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

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

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.

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.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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