 2.6.1: Determine whether each of the equations in 1 through 8 is exact. If...
 2.6.2: Determine whether each of the equations in 1 through 8 is exact. If...
 2.6.3: Determine whether each of the equations in 1 through 8 is exact. If...
 2.6.4: Determine whether each of the equations in 1 through 8 is exact. If...
 2.6.5: Determine whether each of the equations in 1 through 8 is exact. If...
 2.6.6: Determine whether each of the equations in 1 through 8 is exact. If...
 2.6.7: Determine whether each of the equations in 1 through 8 is exact. If...
 2.6.8: Determine whether each of the equations in 1 through 8 is exact. If...
 2.6.9: In each of 9 and 10, solve the given initial value problem and dete...
 2.6.10: In each of 9 and 10, solve the given initial value problem and dete...
 2.6.11: In each of 11 and 12, find the value of b for which the given equat...
 2.6.12: In each of 11 and 12, find the value of b for which the given equat...
 2.6.13: Assume that equation (6) meets the requirements of Theorem 2.6.1 in...
 2.6.14: Show that any separable equation M( x) + N( y) y_ = 0 is also exact.
 2.6.15: In each of 15 and 16, show that the given equation is not exact but...
 2.6.16: In each of 15 and 16, show that the given equation is not exact but...
 2.6.17: Show that if ( Nx My )/M = Q, where Q is a function of y only, then...
 2.6.18: In each of 18 through 21, find an integrating factor and solve the ...
 2.6.19: In each of 18 through 21, find an integrating factor and solve the ...
 2.6.20: In each of 18 through 21, find an integrating factor and solve the ...
 2.6.21: In each of 18 through 21, find an integrating factor and solve the ...
 2.6.22: Solve the differential equation (3xy + y2) + ( x2 + xy) y_ = 0
Solutions for Chapter 2.6: Exact Differential Equations and Integrating Factors
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 2.6: Exact Differential Equations and Integrating Factors
Get Full SolutionsSince 22 problems in chapter 2.6: Exact Differential Equations and Integrating Factors have been answered, more than 13662 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. Chapter 2.6: Exact Differential Equations and Integrating Factors includes 22 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions.

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

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

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

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.