 2.6.1: Determine whether each of the equations in 1 through 12 is exact. I...
 2.6.2: Determine whether each of the equations in 1 through 12 is exact. I...
 2.6.3: Determine whether each of the equations in 1 through 12 is exact. I...
 2.6.4: Determine whether each of the equations in 1 through 12 is exact. I...
 2.6.5: Determine whether each of the equations in 1 through 12 is exact. I...
 2.6.6: Determine whether each of the equations in 1 through 12 is exact. I...
 2.6.7: Determine whether each of the equations in 1 through 12 is exact. I...
 2.6.8: Determine whether each of the equations in 1 through 12 is exact. I...
 2.6.9: Determine whether each of the equations in 1 through 12 is exact. I...
 2.6.10: Determine whether each of the equations in 1 through 12 is exact. I...
 2.6.11: Determine whether each of the equations in 1 through 12 is exact. I...
 2.6.12: Determine whether each of the equations in 1 through 12 is exact. I...
 2.6.13: In each of 13 and 14, solve the given initial value problem and det...
 2.6.14: In each of 13 and 14, solve the given initial value problem and det...
 2.6.15: In each of 15 and 16, find the value of b for which the given equat...
 2.6.16: In each of 15 and 16, find the value of b for which the given equat...
 2.6.17: Assume that Eq. (6) meets the requirements of Theorem 2.6.1 in a re...
 2.6.18: Show that any separable equationM(x) + N(y)y= 0is also exact.In eac...
 2.6.19: x2y3 + x(1 + y2)y= 0, (x, y) = 1/xy3
 2.6.20: . sin yy 2ex sin x+cos y + 2ex cos xyy= 0, (x, y) = yex
 2.6.21: y + (2x yey)y= 0, (x, y) = y
 2.6.22: (x + 2)sin y + (x cos y)y= 0, (x, y) = xex
 2.6.23: Show that if (Nx My)/M = Q, where Q is a function of y only, then t...
 2.6.24: Show that if (Nx My)/(xM yN) = R, where R depends on the quantity x...
 2.6.25: In each of 25 through 31, find an integrating factor and solve the ...
 2.6.26: In each of 25 through 31, find an integrating factor and solve the ...
 2.6.27: In each of 25 through 31, find an integrating factor and solve the ...
 2.6.28: In each of 25 through 31, find an integrating factor and solve the ...
 2.6.29: In each of 25 through 31, find an integrating factor and solve the ...
 2.6.30: In each of 25 through 31, find an integrating factor and solve the ...
 2.6.31: In each of 25 through 31, find an integrating factor and solve the ...
 2.6.32: Solve the differential equation(3xy + y2) + (x2 + xy)y= 0using the ...
Solutions for Chapter 2.6: Exact Equations and Integrating Factors
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 2.6: Exact Equations and Integrating Factors
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. Since 32 problems in chapter 2.6: Exact Equations and Integrating Factors have been answered, more than 16361 students have viewed full stepbystep solutions from this chapter. Chapter 2.6: Exact Equations and Integrating Factors includes 32 full stepbystep solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. This expansive textbook survival guide covers the following chapters and their solutions.

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).

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

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.

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

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.

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

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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 .

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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).

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.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·