 14.2.14.1.14: A string is stretched along the xaxis between (0, 0) and (L, 0). F...
 14.2.14.1.15: Solve the boundaryvalue problem
 14.2.14.1.16: The displacement of a semiinfinite elastic string is determined fr...
 14.2.14.1.17: Solve the boundaryvalue problem in when . Sketch the displacement ...
 14.2.14.1.18: In Example 3 find the displacement u(x, t) when the left end of the...
 14.2.14.1.19: The displacement u(x, t) of a string that is driven by an external ...
 14.2.14.1.20: A uniform bar is clamped at x 0 and is initially at rest. If a cons...
 14.2.14.1.21: A uniform semiinfinite elastic beam moving along the xaxis with a...
 14.2.14.1.22: Solve the boundaryvalue problem
 14.2.14.1.23: Solve the boundaryvalue problem
 14.2.14.1.24: In 1118 use the Laplace transform to solve the heat equation uxx ut...
 14.2.14.1.25: In 1118 use the Laplace transform to solve the heat equation uxx ut...
 14.2.14.1.26: In 1118 use the Laplace transform to solve the heat equation uxx ut...
 14.2.14.1.27: In 1118 use the Laplace transform to solve the heat equation uxx ut...
 14.2.14.1.28: In 1118 use the Laplace transform to solve the heat equation uxx ut...
 14.2.14.1.29: In 1118 use the Laplace transform to solve the heat equation uxx ut...
 14.2.14.1.30: In 1118 use the Laplace transform to solve the heat equation uxx ut...
 14.2.14.1.31: In 1118 use the Laplace transform to solve the heat equation uxx ut...
 14.2.14.1.32: Solve the boundaryvalue problem
 14.2.14.1.33: Show that a solution of the boundaryvalue problem where r is a con...
 14.2.14.1.34: A rod of length L is held at a constant temperature u0 at its ends ...
 14.2.14.1.35: If there is a heat transfer from the lateral surface of a thin wire...
 14.2.14.1.36: A rod of unit length is insulated at x 0 and is kept at temperature...
 14.2.14.1.37: An infinite porous slab of unit width is immersed in a solution of ...
 14.2.14.1.38: A very long telephone transmission line is initially at a constant ...
 14.2.14.1.39: Show that a solution of the boundaryvalue problem
 14.2.14.1.40: In of Exercises 13.3 you were asked to fin the timedependent tempe...
 14.2.14.1.41: Starting at t 0, a concentrated load of magnitude F0 moves with a c...
 14.2.14.1.42: (a) The temperature in a semiinfinite solid is modeled by the boun...
 14.2.14.1.43: (a) In if there is a constant flux of heat into the solid at its le...
 14.2.14.1.44: Humans gather most of our information on the outside world through ...
Solutions for Chapter 14.2: Integral Transforms
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 14.2: Integral Transforms
Get Full SolutionsSince 31 problems in chapter 14.2: Integral Transforms have been answered, more than 23196 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 14.2: Integral Transforms includes 31 full stepbystep solutions. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

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

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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·