 15.2.1: A string is secured to the xaxis at (0, 0) and (L, 0). Find the di...
 15.2.2: Solve the boundaryvalue problem 02 u 0x 2 02 u 0t 2 , 0 , x , 1, t...
 15.2.3: The displacement of a semiinfinite elastic string is determined fr...
 15.2.4: Solve the boundaryvalue problem in when f(t) e sin pt, 0 # t # 1 0...
 15.2.5: In Example 3, find the displacement u(x, t) when the left end of th...
 15.2.6: The displacement u(x, t) of a string that is driven by an external ...
 15.2.7: A uniform bar is clamped at x 0 and is initially at rest. If a cons...
 15.2.8: A uniform semiinfinite elastic beam moving along the xaxis with a...
 15.2.9: Solve the boundaryvalue problem 02 u 0x2 02 u 0t 2 , x . 0, t . 0 ...
 15.2.10: Solve the boundaryvalue problem 02 u 0x2 02 u 0t 2 , x . 0, t . 0 ...
 15.2.11: In 1118, use the Laplace transform to solve the heat equation uxx u...
 15.2.12: In 1118, use the Laplace transform to solve the heat equation uxx u...
 15.2.13: In 1118, use the Laplace transform to solve the heat equation uxx u...
 15.2.14: In 1118, use the Laplace transform to solve the heat equation uxx u...
 15.2.15: In 1118, use the Laplace transform to solve the heat equation uxx u...
 15.2.16: In 1118, use the Laplace transform to solve the heat equation uxx u...
 15.2.17: In 1118, use the Laplace transform to solve the heat equation uxx u...
 15.2.18: In 1118, use the Laplace transform to solve the heat equation uxx u...
 15.2.19: Solve the boundaryvalue problem 02 u 0x 2 0u 0t , q , x , 1, t . 0...
 15.2.20: Show that a solution of the boundaryvalue problem k 02 u 0x 2 r 0u...
 15.2.21: A rod of length L is held at a constant temperature u0 at its ends ...
 15.2.22: If there is a heat transfer from the lateral surface of a thin wire...
 15.2.23: A rod of unit length is insulated at x 0 and is kept at temperature...
 15.2.24: An infinite porous slab of unit width is immersed in a solution of ...
 15.2.25: A very long telephone transmission line is initially at a constant ...
 15.2.26: Starting at t 0, a concentrated load of magnitude F0 moves with a c...
 15.2.27: In of Exercises 14.3 you were asked to find the timedependent temp...
 15.2.28: Show that a solution of the boundaryvalue problem 02 u 0x 2 2 hu 0...
 15.2.29: The temperature in a semiinfinite solid is modeled by the boundary...
 15.2.30: In 29, if there is a constant flux of heat into the solid at its le...
 15.2.31: Use a CAS to obtain the graph of u(x, t) in over the rectangular re...
 15.2.32: Use a CAS to obtain the graph of u(x, t) in over the rectangular re...
 15.2.33: Humans gather most of their information on the outside world throug...
Solutions for Chapter 15.2: Applications of the Laplace Transform
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 15.2: Applications of the Laplace Transform
Get Full SolutionsChapter 15.2: Applications of the Laplace Transform includes 33 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. Since 33 problems in chapter 15.2: Applications of the Laplace Transform have been answered, more than 34438 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6.

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

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

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

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).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

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·

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