 10.5.1: In each of 1 through 6 determine whether the method of separation o...
 10.5.2: In each of 1 through 6 determine whether the method of separation o...
 10.5.3: In each of 1 through 6 determine whether the method of separation o...
 10.5.4: In each of 1 through 6 determine whether the method of separation o...
 10.5.5: In each of 1 through 6 determine whether the method of separation o...
 10.5.6: In each of 1 through 6 determine whether the method of separation o...
 10.5.7: Find the solution of the heat conduction problem 100uxx = ut, 0 < x...
 10.5.8: Find the solution of the heat conduction problem uxx = 4ut, 0 < x <...
 10.5.9: Consider the conduction of heat in a rod 40 cm in length whose ends...
 10.5.10: Consider the conduction of heat in a rod 40 cm in length whose ends...
 10.5.11: Consider the conduction of heat in a rod 40 cm in length whose ends...
 10.5.12: Consider the conduction of heat in a rod 40 cm in length whose ends...
 10.5.13: Consider again the rod in 9. For t = 5 and x = 20 determine how man...
 10.5.14: For the rod in 9: (a) Plot u versus x for t = 5, 10, 20, 40, 100, a...
 10.5.15: Follow the instructions in for the rod in 10.
 10.5.16: Follow the instructions in for the rod in 11.
 10.5.17: For the rod in 12: (a) Plot u versus x for t = 5, 10, 20, 40, 100, ...
 10.5.18: Let a metallic rod 20 cm long be heated to a uniform temperature of...
 10.5.19: For the rod of find the time that will elapse before the center of ...
 10.5.20: In solving differential equations, the computations can almost alwa...
 10.5.21: Consider the equation auxx but + cu = 0, (i) where a, b, and c are ...
 10.5.22: The heat conduction equation in two space dimensions is 2 (uxx + uy...
 10.5.23: The heat conduction equation in two space dimensions may be express...
Solutions for Chapter 10.5: Separation of Variables; Heat Conduction in a Rod
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 10.5: Separation of Variables; Heat Conduction in a Rod
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 10.5: Separation of Variables; Heat Conduction in a Rod includes 23 full stepbystep solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. Since 23 problems in chapter 10.5: Separation of Variables; Heat Conduction in a Rod have been answered, more than 14013 students have viewed full stepbystep solutions from this chapter.

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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

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).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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.

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

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

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

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