- 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
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.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + 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.
Invert A by row operations on [A I] to reach [I A-I].
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)j-I 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.