 2.1.1: In each of 1 through 8: G a. Draw a direction field for the given d...
 2.1.2: In each of 1 through 8: G a. Draw a direction field for the given d...
 2.1.3: In each of 1 through 8: G a. Draw a direction field for the given d...
 2.1.4: In each of 1 through 8: G a. Draw a direction field for the given d...
 2.1.5: In each of 1 through 8: G a. Draw a direction field for the given d...
 2.1.6: In each of 1 through 8: G a. Draw a direction field for the given d...
 2.1.7: In each of 1 through 8: G a. Draw a direction field for the given d...
 2.1.8: In each of 1 through 8: G a. Draw a direction field for the given d...
 2.1.9: In each of 9 through 12, find the solution of the given initial val...
 2.1.10: In each of 9 through 12, find the solution of the given initial val...
 2.1.11: In each of 9 through 12, find the solution of the given initial val...
 2.1.12: In each of 9 through 12, find the solution of the given initial val...
 2.1.13: In each of 13 and 14: G a. Draw a direction field for the given dif...
 2.1.14: In each of 13 and 14: G a. Draw a direction field for the given dif...
 2.1.15: In each of 15 and 16: G a. Draw a direction field for the given dif...
 2.1.16: In each of 15 and 16: G a. Draw a direction field for the given dif...
 2.1.17: Consider the initial value problem y_ + 1 2 y = 2 cos t, y(0) = 1. ...
 2.1.18: Consider the initial value problem y_ + 2 3 y = 1 1 2 t, y(0) = y0....
 2.1.19: Consider the initial value problem y_ + 1 4 y = 3 + 2 cos(2t), y(0)...
 2.1.20: Find the value of y0 for which the solution of the initial value pr...
 2.1.21: Consider the initial value problem y_ 3 2 y = 3t + 2et , y(0) = y0....
 2.1.22: Show that all solutions of 2y_ + ty = 2 [equation (41) of the text]...
 2.1.23: Show that if a and are positive constants, and b is any real number...
 2.1.24: All solutions have the limit 3 as t .
 2.1.25: All solutions are asymptotic to the line y = 3 t as t .
 2.1.26: All solutions are asymptotic to the line y = 2t 5 as t .
 2.1.27: All solutions approach the curve y = 4 t2 as t .
 2.1.28: Variation of Parameters. Consider the following method of solving t...
 2.1.29: In each of 29 and 30, use the method of to solve the given differen...
 2.1.30: In each of 29 and 30, use the method of to solve the given differen...
Solutions for Chapter 2.1: Linear Differential Equations; Method of Integrating Factors
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 2.1: Linear Differential Equations; Method of Integrating Factors
Get Full SolutionsElementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.1: Linear Differential Equations; Method of Integrating Factors includes 30 full stepbystep solutions. Since 30 problems in chapter 2.1: Linear Differential Equations; Method of Integrating Factors have been answered, more than 12382 students have viewed full stepbystep solutions from this chapter.

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

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.