 2.2.1: In each of 1 through 8, solve the given differential equation.y = x2/y
 2.2.2: In each of 1 through 8, solve the given differential equation.y = x...
 2.2.3: In each of 1 through 8, solve the given differential equation.y + y...
 2.2.4: In each of 1 through 8, solve the given differential equation.y = (...
 2.2.5: In each of 1 through 8, solve the given differential equation.y = (...
 2.2.6: In each of 1 through 8, solve the given differential equation.xy = ...
 2.2.7: In each of 1 through 8, solve the given differential equation.dydx ...
 2.2.8: In each of 1 through 8, solve the given differential equation.dydx ...
 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: 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  10th Edition
ISBN: 9780470458327
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 12175 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327. Chapter 2.2: Separable Equations includes 38 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Column space C (A) =
space of all combinations of the columns of A.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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

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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.