 14.4.14.1.66: Use the Fourier integral transforms of this section to solve the gi...
 14.4.14.1.67: Use the Fourier integral transforms of this section to solve the gi...
 14.4.14.1.68: Find the temperature u(x, t) in a semiinfinite rod if u(0, t) u0, ...
 14.4.14.1.69: Use the result , to show that the solution of can be written as
 14.4.14.1.70: Find the temperature u(x, t) in a semiinfinite rod if u(0, t) 0, t...
 14.4.14.1.71: Solve if the condition at the left boundary is , where A is a const...
 14.4.14.1.72: Solve if the end x 0 is insulated.
 14.4.14.1.73: ind the temperature u(x, t) in a semiinfinite rod if u(0, t) 1, t ...
 14.4.14.1.74: (a) (b) If g(x) 0, show that the solution of part (a) can be writte...
 14.4.14.1.75: Find the displacement u(x, t) of a semiinfinite string if
 14.4.14.1.76: Solve the problem in Example 2 if the boundary conditions at x 0 an...
 14.4.14.1.77: Solve the problem in Example 2 if the boundary condition at y 0 is ...
 14.4.14.1.78: Find the steadystate temperature u(x, y) in a plate defined by x 0...
 14.4.14.1.79: Solve if the boundary condition at x 0 is u(0, y) 0, y 0.
 14.4.14.1.80: Use the Fourier integral transforms of this section to solve the gi...
 14.4.14.1.81: Use the Fourier integral transforms of this section to solve the gi...
 14.4.14.1.82: In 17 and 18 find the steadystate temperature in the plate given i...
 14.4.14.1.83: In 17 and 18 find the steadystate temperature in the plate given i...
 14.4.14.1.84: Use the result to solve the boundaryvalue problem
 14.4.14.1.85: If , then the convolution theorem for the Fourier transform is give...
 14.4.14.1.86: Use the transform given in to find the steadystate temperature in ...
 14.4.14.1.87: The solution of can be integrated. Use entries 42 and 43 of the tab...
 14.4.14.1.88: Use the solution given in to rewrite the solution of Example 1 in a...
 14.4.14.1.89: The steadystate temperatures in a semiinfinit cylinder are descri...
 14.4.14.1.90: Find the steadystate temperatures in the semiinfinite cylinder in ...
 14.4.14.1.91: (a) Suppose where Find f(x). (b) Use part (a) to show that
 14.4.14.1.92: Assume that u0 100 and k 1 in the solution in 23. Use a CAS to grap...
Solutions for Chapter 14.4: Integral Transforms
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 14.4: Integral Transforms
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Since 27 problems in chapter 14.4: Integral Transforms have been answered, more than 23273 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. Chapter 14.4: Integral Transforms includes 27 full stepbystep 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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

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

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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 •

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).