 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 dete...
 2.6.14: In each of 13 and 14 solve the given initial value problem and dete...
 2.6.15: In each of 15 and 16 find the value of b for which the given equati...
 2.6.16: In each of 15 and 16 find the value of b for which the given equati...
 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 equation M(x) + N(y)y = 0 is also exact.
 2.6.19: In each of 19 through 22 show that the given equation is not exact ...
 2.6.20: In each of 19 through 22 show that the given equation is not exact ...
 2.6.21: In each of 19 through 22 show that the given equation is not exact ...
 2.6.22: In each of 19 through 22 show that the given equation is not exact ...
 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 g...
 2.6.26: In each of 25 through 31 find an integrating factor and solve the g...
 2.6.27: In each of 25 through 31 find an integrating factor and solve the g...
 2.6.28: In each of 25 through 31 find an integrating factor and solve the g...
 2.6.29: In each of 25 through 31 find an integrating factor and solve the g...
 2.6.30: In each of 25 through 31 find an integrating factor and solve the g...
 2.6.31: In each of 25 through 31 find an integrating factor and solve the g...
 2.6.32: Solve the differential equation (3xy + y2 ) + (x2 + xy)y = 0 using ...
Solutions for Chapter 2.6: Exact Equations and Integrating Factors
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 2.6: Exact Equations and Integrating Factors
Get Full SolutionsChapter 2.6: Exact Equations and Integrating Factors includes 32 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. Since 32 problems in chapter 2.6: Exact Equations and Integrating Factors have been answered, more than 14395 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

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.

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

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.

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

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

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

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

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

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

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.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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