 2.2.1: In each of 1 through 8, solve the given differential equationy= x2/y
 2.2.2: In each of 1 through 8, solve the given differential equationy= x2/...
 2.2.3: In each of 1 through 8, solve the given differential equationy+ y2 ...
 2.2.4: In each of 1 through 8, solve the given differential equationy= (3x...
 2.2.5: In each of 1 through 8, solve the given differential equationy= (co...
 2.2.6: In each of 1 through 8, solve the given differential equationxy= (1...
 2.2.7: In each of 1 through 8, solve the given differential equationdydx =...
 2.2.8: In each of 1 through 8, solve the given differential equationdydx =...
 2.2.9: In each of 9 through 20:(a) Find the solution of the given initial ...
 2.2.10: In each of 9 through 20:(a) Find the solution of the given initial ...
 2.2.11: In each of 9 through 20:(a) Find the solution of the given initial ...
 2.2.12: In each of 9 through 20:(a) Find the solution of the given initial ...
 2.2.13: In each of 9 through 20:(a) Find the solution of the given initial ...
 2.2.14: In each of 9 through 20:(a) Find the solution of the given initial ...
 2.2.15: In each of 9 through 20:(a) Find the solution of the given initial ...
 2.2.16: In each of 9 through 20:(a) Find the solution of the given initial ...
 2.2.17: In each of 9 through 20:(a) Find the solution of the given initial ...
 2.2.18: In each of 9 through 20:(a) Find the solution of the given initial ...
 2.2.19: In each of 9 through 20:(a) Find the solution of the given initial ...
 2.2.20: In each of 9 through 20:(a) Find the solution of the given initial ...
 2.2.21: Some of the results requested in 21 through 28 can be obtained eith...
 2.2.22: Some of the results requested in 21 through 28 can be obtained eith...
 2.2.23: Some of the results requested in 21 through 28 can be obtained eith...
 2.2.24: Some of the results requested in 21 through 28 can be obtained eith...
 2.2.25: Some of the results requested in 21 through 28 can be obtained eith...
 2.2.26: Some of the results requested in 21 through 28 can be obtained eith...
 2.2.27: Some of the results requested in 21 through 28 can be obtained eith...
 2.2.28: Some of the results requested in 21 through 28 can be obtained eith...
 2.2.29: Solve the equationdydx = ay + bcy + d ,where a, b, c, and d are con...
 2.2.30: Consider the equationdydx = y 4xx y . (i)(a) Show that Eq. (i) can ...
 2.2.31: dydx = x2 + xy + y2x
 2.2.32: dydx = x2 + 3y22xy
 2.2.33: dydx = 4y 3x2x y
 2.2.34: dydx = 4x + 3y2x + y
 2.2.35: dydx = x + 3yx y
 2.2.36: (x2 + 3xy + y2) dx x2 dy = 0
 2.2.37: dydx = x2 3y22xy
 2.2.38: dydx = 3y2 x22xy
Solutions for Chapter 2.2: Separable Equations
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 2.2: Separable Equations
Get Full SolutionsSince 38 problems in chapter 2.2: Separable Equations have been answered, more than 16341 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: 9780470458310. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.2: Separable Equations includes 38 full stepbystep 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).

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

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.

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.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

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 IIĀ·

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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