 1.9.1: For 14, determine whether the given differential equation is exact....
 1.9.2: For 14, determine whether the given differential equation is exact....
 1.9.3: For 14, determine whether the given differential equation is exact.
 1.9.4: For 14, determine whether the given differential equation is exact....
 1.9.5: For 515, solve the given differential equation.2xy dx + (x2 + 1) dy...
 1.9.6: For 515, solve the given differential equation.
 1.9.7: For 515, solve the given differential equation.(4e2x + 2xy y2) dx +...
 1.9.8: For 515, solve the given differential equation.
 1.9.9: For 515, solve the given differential equation.
 1.9.10: For 515, solve the given differential equation.(2y2e2x + 3x2) dx + ...
 1.9.11: For 515, solve the given differential equation.
 1.9.12: For 515, solve the given differential equation.
 1.9.13: For 515, solve the given differential equation.[1 + ln (xy)] dx + x...
 1.9.14: For 515, solve the given differential equation.
 1.9.15: For 515, solve the given differential equation.(2xy + cos y) dx + (...
 1.9.16: For 1618, solve the given initialvalue problem.2x2 y + 4xy = 3 sin...
 1.9.17: For 1618, solve the given initialvalue problem.
 1.9.18: For 1618, solve the given initialvalue problem.(yexy + cos x) dx +...
 1.9.19: Show that if (x, y) is a potential function for M(x, y) dx + N(x, y...
 1.9.20: For 2022, determine whether the given function is an integrating fa...
 1.9.21: For 2022, determine whether the given function is an integrating fa...
 1.9.22: For 2022, determine whether the given function is an integrating fa...
 1.9.23: For 2329, determine an integrating factor for the given differentia...
 1.9.24: For 2329, determine an integrating factor for the given differentia...
 1.9.25: For 2329, determine an integrating factor for the given differentia...
 1.9.26: For 2329, determine an integrating factor for the given differentia...
 1.9.27: For 2329, determine an integrating factor for the given differentia...
 1.9.28: For 2329, determine an integrating factor for the given differentia...
 1.9.29: For 2329, determine an integrating factor for the given differentia...
 1.9.30: For 3032, determine the values of the constants r and s such that I...
 1.9.31: For 3032, determine the values of the constants r and s such that I...
 1.9.32: For 3032, determine the values of the constants r and s such that I...
 1.9.33: Prove that if (My Nx )/M = g(y), a function of y only, then an inte...
 1.9.34: Prove that if (My Nx )/M = g(y), a function of y only, then an inte...
Solutions for Chapter 1.9: Exact Differential Equations
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 1.9: Exact Differential Equations
Get Full SolutionsSince 34 problems in chapter 1.9: Exact Differential Equations have been answered, more than 21272 students have viewed full stepbystep solutions from this chapter. Differential Equations was written by and is associated to the ISBN: 9780321964670. Chapter 1.9: Exact Differential Equations includes 34 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations, edition: 4.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

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

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

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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 addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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.