 10.5.1: In each of 1 through 6, determine whether the method of separation ...
 10.5.2: In each of 1 through 6, determine whether the method of separation ...
 10.5.3: In each of 1 through 6, determine whether the method of separation ...
 10.5.4: In each of 1 through 6, determine whether the method of separation ...
 10.5.5: In each of 1 through 6, determine whether the method of separation ...
 10.5.6: In each of 1 through 6, determine whether the method of separation ...
 10.5.7: Find the solution of the heat conduction problem100uxx = ut, 0 < x ...
 10.5.8: Find the solution of the heat conduction problemuxx = 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 ma...
 10.5.14: For the rod in 9:(a) Plot u versus x for t = 5, 10, 20, 40, 100, an...
 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, a...
 10.5.18: Let a metallic rod 20 cm long be heated to a uniform temperature of...
 10.5.19: For the rod of 18, find the time that will elapse before the center...
 10.5.20: In solving differential equations, the computations can almost alwa...
 10.5.21: Consider the equationauxx but + cu = 0, (i)where a, b, and c are co...
 10.5.22: The heat conduction equation in two space dimensions is2(uxx + uyy)...
 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  10th Edition
ISBN: 9780470458310
Solutions for Chapter 10.5: Separation of Variables; Heat Conduction in a Rod
Get Full SolutionsChapter 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: 9780470458310. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. Since 23 problems in chapter 10.5: Separation of Variables; Heat Conduction in a Rod have been answered, more than 16837 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

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.)

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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

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].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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 .

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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.

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