 2.3.2.3.1: For the following partial differential equations, what ordinary dif...
 2.3.2.3.2: Consider the differential equation d2 dx2 + = 0. Determine the eige...
 2.3.2.3.3: Consider the differential equation d2 dx2 + = 0. Determine the eige...
 2.3.2.3.4: Consider u t = k 2u x2 , subject to u(0, t)=0, u(L, t) = 0, and u(x...
 2.3.2.3.5: Evaluate (be careful if n = m) L 0 sin nx L sin mx L dx for n > 0,m...
 2.3.2.3.6: Evaluate L 0 cos nx L cos mx L dx for n 0, m 0. Use the trigonometr...
 2.3.2.3.7: Consider the following boundary value problem (if necessary, see Se...
 2.3.2.3.8: Consider u t = k 2u x2 u. This corresponds to a onedimensional rod...
 2.3.2.3.9: Redo Exercise 2.3.8 if < 0. [Be especially careful if /k = (n/L) 2.]
 2.3.2.3.10: For two and threedimensional vectors, the fundamental property of...
 2.3.2.3.11: Solve the heat equation u t = k 2u x2 subject to the following cond...
Solutions for Chapter 2.3: Method of Separation of Variables
Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems  5th Edition
ISBN: 9780321797056
Solutions for Chapter 2.3: Method of Separation of Variables
Get Full SolutionsThis textbook survival guide was created for the textbook: Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, edition: 5. Chapter 2.3: Method of Separation of Variables includes 11 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 11 problems in chapter 2.3: Method of Separation of Variables have been answered, more than 7738 students have viewed full stepbystep solutions from this chapter. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems was written by and is associated to the ISBN: 9780321797056.

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.

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.