 APPENDIX A.1: dy dt = y 4t + y2 8yt + 16t 2 + 4, let u = y 4t
 APPENDIX A.2: dy dt = y2 + t y y2 + 3t2 , let u = y t
 APPENDIX A.3: dy dt = t(y + t y2) + cost y, let u = t y
 APPENDIX A.4: dy dt = ey + t2 ey , let u = ey
 APPENDIX A.5: dy dt = (y t) 2 (y t) 1
 APPENDIX A.6: dy dt = y2 t + 2y 4t + y t
 APPENDIX A.7: dy dt = y cost y y t
 APPENDIX A.8: dy dt = t y 2 + et2/2 2y , let y = u
 APPENDIX A.9: dy dt = y 1 + t y t + t 2(1 + t), let u = y 1 + t
 APPENDIX A.10: dy dt = y2 2yt + t 2 + y t + 1, let u = y t
 APPENDIX A.11: y = u + t
 APPENDIX A.12: y = u
 APPENDIX A.13: y = u2
 APPENDIX A.14: Consider a 20gallon vat that at time t = 0 contains 5 gallons of c...
 APPENDIX A.15: Consider a very large vat that initially contains 10 gallons of cle...
 APPENDIX A.16: Given a differential equation of the form dy dt = g y t , show that...
 APPENDIX A.17: dy dt = 10y3 1
 APPENDIX A.18: dy dt = (y + 1)(y 3)
 APPENDIX A.19: dy dt = (y + 1)(3 y)
 APPENDIX A.20: dy dt = y3 3y2 + y
 APPENDIX A.21: Consider the differential equation dy/dt = f (y), where f (y) is a ...
 APPENDIX A.22: So far we have changed only the dependent variable. It is also poss...
 APPENDIX A.23: dy dt = y + y3
 APPENDIX A.24: dy dt = y + t y3
 APPENDIX A.25: dy dt = 1 t y y4
 APPENDIX A.26: dy dt = y + y50
 APPENDIX A.27: dy dt = (2t + 1/t)y y2 t 2, y1(t) = t
 APPENDIX A.28: dy dt = (t 4 t 2 + 2t) + (1 2t 2)y + y2, y1(t) = t2
 APPENDIX A.29: dy dt = 2 sin t + cost + t 2 sin2 t 2(1 + t 2 sin t)y + t 2 y2, y1(...
 APPENDIX A.30: Suppose we must solve a Bernoulli equation dy dt = r(t)y + a(t)yn, ...
 APPENDIX A.31: In the discussion of the Riccati equation dy/dt = r(t)+a(t)y +b(t)y...
 APPENDIX A.32: Construct your own Bernoulli equation exercise of the form dy/dt = ...
 APPENDIX A.33: Construct your own Riccati equation exercise from the function y1(t...
Solutions for Chapter APPENDIX A: CHANGING VARIABLES
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter APPENDIX A: CHANGING VARIABLES
Get Full SolutionsChapter APPENDIX A: CHANGING VARIABLES includes 33 full stepbystep solutions. Differential Equations 00 was written by and is associated to the ISBN: 9780495561989. Since 33 problems in chapter APPENDIX A: CHANGING VARIABLES have been answered, more than 16292 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations 00, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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!).

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(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 .

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

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.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

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