 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 equation
 2.2.5: In each of 1 through 8 solve the given differential equation
 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 equation dy dx = ay + b cy + d , where a, b, c, and d are...
 2.2.30: Consider the equation dy dx = y 4x x y . (i) (a) Show that Eq. (i) ...
 2.2.31: The method outlined in can be used for any homogeneous equation. Th...
 2.2.32: The method outlined in can be used for any homogeneous equation. Th...
 2.2.33: The method outlined in can be used for any homogeneous equation. Th...
 2.2.34: The method outlined in can be used for any homogeneous equation. Th...
 2.2.35: The method outlined in can be used for any homogeneous equation. Th...
 2.2.36: The method outlined in can be used for any homogeneous equation. Th...
 2.2.37: The method outlined in can be used for any homogeneous equation. Th...
 2.2.38: The method outlined in can be used for any homogeneous equation. Th...
Solutions for Chapter 2.2: Separable Equations
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 2.2: Separable Equations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 38 problems in chapter 2.2: Separable Equations have been answered, more than 12713 students have viewed full stepbystep solutions from this chapter. 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. Chapter 2.2: Separable Equations includes 38 full stepbystep solutions.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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).

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.

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

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).

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

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.