 2.1.1: In each of 1 through 12:(a) Draw a direction field for the given di...
 2.1.2: In each of 1 through 12:(a) Draw a direction field for the given di...
 2.1.3: In each of 1 through 12:(a) Draw a direction field for the given di...
 2.1.4: In each of 1 through 12:(a) Draw a direction field for the given di...
 2.1.5: In each of 1 through 12:(a) Draw a direction field for the given di...
 2.1.6: In each of 1 through 12:(a) Draw a direction field for the given di...
 2.1.7: In each of 1 through 12:(a) Draw a direction field for the given di...
 2.1.8: In each of 1 through 12:(a) Draw a direction field for the given di...
 2.1.9: In each of 1 through 12:(a) Draw a direction field for the given di...
 2.1.10: In each of 1 through 12:(a) Draw a direction field for the given di...
 2.1.11: In each of 1 through 12:(a) Draw a direction field for the given di...
 2.1.12: In each of 1 through 12:(a) Draw a direction field for the given di...
 2.1.13: In each of 13 through 20, find the solution of the given initial va...
 2.1.14: In each of 13 through 20, find the solution of the given initial va...
 2.1.15: In each of 13 through 20, find the solution of the given initial va...
 2.1.16: In each of 13 through 20, find the solution of the given initial va...
 2.1.17: In each of 13 through 20, find the solution of the given initial va...
 2.1.18: In each of 13 through 20, find the solution of the given initial va...
 2.1.19: In each of 13 through 20, find the solution of the given initial va...
 2.1.20: In each of 13 through 20, find the solution of the given initial va...
 2.1.21: In each of 21 through 23:(a) Draw a direction field for the given d...
 2.1.22: In each of 21 through 23:(a) Draw a direction field for the given d...
 2.1.23: In each of 21 through 23:(a) Draw a direction field for the given d...
 2.1.24: In each of 24 through 26:(a) Draw a direction field for the given d...
 2.1.25: In each of 24 through 26:(a) Draw a direction field for the given d...
 2.1.26: In each of 24 through 26:(a) Draw a direction field for the given d...
 2.1.27: Consider the initial value problemy+ 12 y = 2 cost, y(0) = 1.Find t...
 2.1.28: Consider the initial value problemy+ 23 y = 1 12 t, y(0) = y0.Find ...
 2.1.29: Consider the initial value problemy+ 14 y = 3 + 2 cos 2t, y(0) = 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 problemy 32 y = 3t + 2et, y(0) = y0.Find...
 2.1.32: Show that all solutions of 2y+ ty = 2 [Eq. (41) of the text] approa...
 2.1.33: Show that if a and are positive constants, and b is any real number...
 2.1.34: All solutions have the limit 3 as t .
 2.1.35: All solutions are asymptotic to the line y = 3 t as t .
 2.1.36: All solutions are asymptotic to the line y = 2t 5 as t .
 2.1.37: All solutions approach the curve y = 4 t 2 as t .
 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 diff...
 2.1.40: In each of 39 through 42, use the method of to solve the given diff...
 2.1.41: In each of 39 through 42, use the method of to solve the given diff...
 2.1.42: In each of 39 through 42, use the method of to solve the given diff...
Solutions for Chapter 2.1: Linear Equations; Method of Integrating Factors
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 2.1: Linear Equations; Method of Integrating Factors
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. Since 42 problems in chapter 2.1: Linear Equations; Method of Integrating Factors have been answered, more than 16437 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.1: Linear Equations; Method of Integrating Factors includes 42 full stepbystep solutions.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

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

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.

Outer product uv T
= column times row = rank one matrix.

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.