 2.1.1: In each of 1 through 12: (a) Draw a direction field for the given d...
 2.1.2: In each of 1 through 12: (a) Draw a direction field for the given d...
 2.1.3: In each of 1 through 12: (a) Draw a direction field for the given d...
 2.1.4: In each of 1 through 12: (a) Draw a direction field for the given d...
 2.1.5: In each of 1 through 12: (a) Draw a direction field for the given d...
 2.1.6: In each of 1 through 12: (a) Draw a direction field for the given d...
 2.1.7: In each of 1 through 12: (a) Draw a direction field for the given d...
 2.1.8: In each of 1 through 12: (a) Draw a direction field for the given d...
 2.1.9: In each of 1 through 12: (a) Draw a direction field for the given d...
 2.1.10: In each of 1 through 12: (a) Draw a direction field for the given d...
 2.1.11: In each of 1 through 12: (a) Draw a direction field for the given d...
 2.1.12: In each of 1 through 12: (a) Draw a direction field for the given d...
 2.1.13: In each of 13 through 20 find the solution of the given initial val...
 2.1.14: In each of 13 through 20 find the solution of the given initial val...
 2.1.15: In each of 13 through 20 find the solution of the given initial val...
 2.1.16: In each of 13 through 20 find the solution of the given initial val...
 2.1.17: In each of 13 through 20 find the solution of the given initial val...
 2.1.18: In each of 13 through 20 find the solution of the given initial val...
 2.1.19: In each of 13 through 20 find the solution of the given initial val...
 2.1.20: In each of 13 through 20 find the solution of the given initial val...
 2.1.21: In each of 21 through 23: (a) Draw a direction field for the given ...
 2.1.22: In each of 21 through 23: (a) Draw a direction field for the given ...
 2.1.23: In each of 21 through 23: (a) Draw a direction field for the given ...
 2.1.24: In each of 24 through 26: (a) Draw a direction field for the given ...
 2.1.25: In each of 24 through 26: (a) Draw a direction field for the given ...
 2.1.26: In each of 24 through 26: (a) Draw a direction field for the given ...
 2.1.27: Consider the initial value problem y + 1 2 y = 2 cost, y(0) = 1. Fi...
 2.1.28: Consider the initial value problem y + 2 3 y = 1 1 2 t, y(0) = y0. ...
 2.1.29: Consider the initial value problem y + 1 4 y = 3 + 2 cos 2t, y(0) =...
 2.1.30: Find the value of y0 for which the solution of the initial value pr...
 2.1.31: Consider the initial value problem y 3 2 y = 3t + 2et , y(0) = y0.F...
 2.1.32: Show that all solutions of 2y + ty = 2 [Eq. (41) of the text] appro...
 2.1.33: Show that if a and are positive constants, and b is any real number...
 2.1.34: In each of 34 through 37 construct a first order linear differentia...
 2.1.35: In each of 34 through 37 construct a first order linear differentia...
 2.1.36: In each of 34 through 37 construct a first order linear differentia...
 2.1.37: In each of 34 through 37 construct a first order linear differentia...
 2.1.38: Variation of Parameters. Consider the following method of solving t...
 2.1.39: In each of 39 through 42 use the method of to solve the given diffe...
 2.1.40: In each of 39 through 42 use the method of to solve the given diffe...
 2.1.41: In each of 39 through 42 use the method of to solve the given diffe...
 2.1.42: In each of 39 through 42 use the method of to solve the given diffe...
Solutions for Chapter 2.1: Linear Equations; Method of Integrating Factors
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 2.1: Linear Equations; Method of Integrating Factors
Get Full SolutionsChapter 2.1: Linear Equations; Method of Integrating Factors includes 42 full stepbystep solutions. Since 42 problems in chapter 2.1: Linear Equations; Method of Integrating Factors have been answered, more than 14239 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. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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

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

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.

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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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.

Solvable system Ax = b.
The right side b is in the column space of A.

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