 7.4.7.1.186: In 18 use Theorem 7.4.1 to evaluate the given Laplace transform.{te...
 7.4.7.1.187: In 18 use Theorem 7.4.1 to evaluate the given Laplace transform.{t ...
 7.4.7.1.188: In 18 use Theorem 7.4.1 to evaluate the given Laplace transform.{t ...
 7.4.7.1.189: In 18 use Theorem 7.4.1 to evaluate the given Laplace transform.{t ...
 7.4.7.1.190: In 18 use Theorem 7.4.1 to evaluate the given Laplace transform.{t ...
 7.4.7.1.191: In 18 use Theorem 7.4.1 to evaluate the given Laplace transform.{t ...
 7.4.7.1.192: In 18 use Theorem 7.4.1 to evaluate the given Laplace transform.{te...
 7.4.7.1.193: In 18 use Theorem 7.4.1 to evaluate the given Laplace transform.{te...
 7.4.7.1.194: In 914 use the Laplace transform to solve the given initialvalue p...
 7.4.7.1.195: In 914 use the Laplace transform to solve the given initialvalue p...
 7.4.7.1.196: In 914 use the Laplace transform to solve the given initialvalue p...
 7.4.7.1.197: In 914 use the Laplace transform to solve the given initialvalue p...
 7.4.7.1.198: In 914 use the Laplace transform to solve the given initialvalue p...
 7.4.7.1.199: In 914 use the Laplace transform to solve the given initialvalue p...
 7.4.7.1.200: In 15 and 16 use a graphing utility to graph the indicated solution...
 7.4.7.1.201: In 15 and 16 use a graphing utility to graph the indicated solution...
 7.4.7.1.202: In some instances the Laplace transform can be used to solve linear...
 7.4.7.1.203: In some instances the Laplace transform can be used to solve linear...
 7.4.7.1.204: In 1930 use Theorem 7.4.2 to evaluate the given Laplace transform. ...
 7.4.7.1.205: In 1930 use Theorem 7.4.2 to evaluate the given Laplace transform. ...
 7.4.7.1.206: In 1930 use Theorem 7.4.2 to evaluate the given Laplace transform. ...
 7.4.7.1.207: In 1930 use Theorem 7.4.2 to evaluate the given Laplace transform. ...
 7.4.7.1.208: In 1930 use Theorem 7.4.2 to evaluate the given Laplace transform. ...
 7.4.7.1.209: In 1930 use Theorem 7.4.2 to evaluate the given Laplace transform. ...
 7.4.7.1.210: In 1930 use Theorem 7.4.2 to evaluate the given Laplace transform. ...
 7.4.7.1.211: In 1930 use Theorem 7.4.2 to evaluate the given Laplace transform. ...
 7.4.7.1.212: In 1930 use Theorem 7.4.2 to evaluate the given Laplace transform. ...
 7.4.7.1.213: In 1930 use Theorem 7.4.2 to evaluate the given Laplace transform. ...
 7.4.7.1.214: In 1930 use Theorem 7.4.2 to evaluate the given Laplace transform. ...
 7.4.7.1.215: In 1930 use Theorem 7.4.2 to evaluate the given Laplace transform. ...
 7.4.7.1.216: In 3134 use (8) to evaluate the given inverse transform. 1 1 s(s 1)
 7.4.7.1.217: In 3134 use (8) to evaluate the given inverse transform. 1 1 s2 (s 1)
 7.4.7.1.218: In 3134 use (8) to evaluate the given inverse transform.2 1 1 s3 (s 1)
 7.4.7.1.219: In 3134 use (8) to evaluate the given inverse transform. 1 1 s(s a) 2
 7.4.7.1.220: The table in Appendix III does not contain an entry for . (a) Use (...
 7.4.7.1.221: Use the Laplace transform and the results of to solve the initialv...
 7.4.7.1.222: In 3746 use the Laplace transform to solve the given integral equat...
 7.4.7.1.223: In 3746 use the Laplace transform to solve the given integral equat...
 7.4.7.1.224: In 3746 use the Laplace transform to solve the given integral equat...
 7.4.7.1.225: In 3746 use the Laplace transform to solve the given integral equat...
 7.4.7.1.226: In 3746 use the Laplace transform to solve the given integral equat...
 7.4.7.1.227: In 3746 use the Laplace transform to solve the given integral equat...
 7.4.7.1.228: In 3746 use the Laplace transform to solve the given integral equat...
 7.4.7.1.229: In 3746 use the Laplace transform to solve the given integral equat...
 7.4.7.1.230: In 3746 use the Laplace transform to solve the given integral equat...
 7.4.7.1.231: In 3746 use the Laplace transform to solve the given integral equat...
 7.4.7.1.232: In 47 and 48 solve equation (10) subject to i(0) 0 with L, R, C, an...
 7.4.7.1.233: In 47 and 48 solve equation (10) subject to i(0) 0 with L, R, C, an...
 7.4.7.1.234: In 4954 use Theorem 7.4.3 to find the Laplace transform of the give...
 7.4.7.1.235: In 4954 use Theorem 7.4.3 to find the Laplace transform of the give...
 7.4.7.1.236: In 4954 use Theorem 7.4.3 to find the Laplace transform of the give...
 7.4.7.1.237: In 4954 use Theorem 7.4.3 to find the Laplace transform of the give...
 7.4.7.1.238: In 4954 use Theorem 7.4.3 to find the Laplace transform of the give...
 7.4.7.1.239: In 4954 use Theorem 7.4.3 to find the Laplace transform of the give...
 7.4.7.1.240: In 55 and 56 solve equation (15) subject to i(0) 0 with E(t) as giv...
 7.4.7.1.241: In 55 and 56 solve equation (15) subject to i(0) 0 with E(t) as giv...
 7.4.7.1.242: In 57 and 58 solve the model for a driven spring/ mass system with ...
 7.4.7.1.243: In 57 and 58 solve the model for a driven spring/ mass system with ...
 7.4.7.1.244: Discuss how Theorem 7.4.1 can be used to fin
 7.4.7.1.245: In Section 6.4 we saw that ty y ty 0 is Bessels equation of order n...
 7.4.7.1.246: (a) Laguerres differential equation ty (1 t)y ny 0 is known to poss...
 7.4.7.1.247: The Laplace transform exists, but without find ing it solve the ini...
 7.4.7.1.248: Solve the integral equation f(t) et et t 0 e t f() d. y
 7.4.7.1.249: (a) Show that the square wave function E(t) given in Figure 7.4.4 c...
 7.4.7.1.250: Use the Laplace transform as an aide in evaluating the improper int...
 7.4.7.1.251: If we assume that exists and , then Use this result to find the Lap...
 7.4.7.1.252: Transform of the Logarithm Because has an infinite discontinuity at...
 7.4.7.1.253: In this problem you are led through the commands in Mathematica tha...
 7.4.7.1.254: Appropriately modify the procedure of to find a solution o
 7.4.7.1.255: The charge q(t) on a capacitor in an LCseries circuit is given by ...
Solutions for Chapter 7.4: The Laplace Transform
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 7.4: The Laplace Transform
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Chapter 7.4: The Laplace Transform includes 70 full stepbystep solutions. Since 70 problems in chapter 7.4: The Laplace Transform have been answered, more than 20121 students have viewed full stepbystep solutions from this chapter. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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

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

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.

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

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.