 2.5.1: Each DE in 114 is homogeneous. In 110, solve the given differential...
 2.5.2: Each DE in 114 is homogeneous. In 110, solve the given differential...
 2.5.3: Each DE in 114 is homogeneous. In 110, solve the given differential...
 2.5.4: Each DE in 114 is homogeneous. In 110, solve the given differential...
 2.5.5: Each DE in 114 is homogeneous. In 110, solve the given differential...
 2.5.6: Each DE in 114 is homogeneous. In 110, solve the given differential...
 2.5.7: Each DE in 114 is homogeneous. In 110, solve the given differential...
 2.5.8: Each DE in 114 is homogeneous. In 110, solve the given differential...
 2.5.9: Each DE in 114 is homogeneous. In 110, solve the given differential...
 2.5.10: Each DE in 114 is homogeneous. In 110, solve the given differential...
 2.5.11: Each DE in 114 is homogeneous. In 1114, solve the given initialval...
 2.5.12: Each DE in 114 is homogeneous. In 1114, solve the given initialval...
 2.5.13: Each DE in 114 is homogeneous. In 1114, solve the given initialval...
 2.5.14: Each DE in 114 is homogeneous. In 1114, solve the given initialval...
 2.5.15: Each DE in 1522 is a Bernoulli equation. In 1520, solve the given d...
 2.5.16: Each DE in 1522 is a Bernoulli equation. In 1520, solve the given d...
 2.5.17: Each DE in 1522 is a Bernoulli equation. In 1520, solve the given d...
 2.5.18: Each DE in 1522 is a Bernoulli equation. In 1520, solve the given d...
 2.5.19: Each DE in 1522 is a Bernoulli equation. In 1520, solve the given d...
 2.5.20: Each DE in 1522 is a Bernoulli equation. In 1520, solve the given d...
 2.5.21: Each DE in 1522 is a Bernoulli equation. In 21 and 22, solve the gi...
 2.5.22: Each DE in 1522 is a Bernoulli equation. In 21 and 22, solve the gi...
 2.5.23: Each DE in 2330 is of the form given in (5). In 2328, solve the giv...
 2.5.24: Each DE in 2330 is of the form given in (5). In 2328, solve the giv...
 2.5.25: Each DE in 2330 is of the form given in (5). In 2328, solve the giv...
 2.5.26: Each DE in 2330 is of the form given in (5). In 2328, solve the giv...
 2.5.27: Each DE in 2330 is of the form given in (5). In 2328, solve the giv...
 2.5.28: Each DE in 2330 is of the form given in (5). In 2328, solve the giv...
 2.5.29: Each DE in 2330 is of the form given in (5). In 29 and 30, solve th...
 2.5.30: Each DE in 2330 is of the form given in (5). In 29 and 30, solve th...
 2.5.31: Explain why it is always possible to express any homogeneous differ...
 2.5.32: Put the homogeneous differential equation (5x2 2y2 ) dx xy dy 0 int...
 2.5.33: (a) Determine two singular solutions of the DE in 10. (b) If the in...
 2.5.34: In Example 3, the solution y(x) becomes unbounded as x S q. Neverth...
 2.5.35: The differential equation dy dx P(x) Q(x)y R(x)y2 is known as Ricca...
 2.5.36: Devise an appropriate substitution to solve xy y ln(xy).
 2.5.37: Population Growth In the study of population dynamics one of the mo...
Solutions for Chapter 2.5: Solutions by Substitutions
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 2.5: Solutions by Substitutions
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. This expansive textbook survival guide covers the following chapters and their solutions. Since 37 problems in chapter 2.5: Solutions by Substitutions have been answered, more than 39487 students have viewed full stepbystep solutions from this chapter. Chapter 2.5: Solutions by Substitutions includes 37 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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.

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.