 7.1: In 1 and 2, use the definition of the Laplace transform to determine
 7.2: In 1 and 2, use the definition of the Laplace transform to determine
 7.3: In 310, determine the Laplace transform of the given function.
 7.4: In 310, determine the Laplace transform of the given function.
 7.5: In 310, determine the Laplace transform of the given function.
 7.6: In 310, determine the Laplace transform of the given function.
 7.7: In 310, determine the Laplace transform of the given function.
 7.8: In 310, determine the Laplace transform of the given function.
 7.9: In 310, determine the Laplace transform of the given function.
 7.10: In 310, determine the Laplace transform of the given function.
 7.11: In 1117, determine the inverse Laplace transform of the given funct...
 7.12: In 1117, determine the inverse Laplace transform of the given funct...
 7.13: In 1117, determine the inverse Laplace transform of the given funct...
 7.14: In 1117, determine the inverse Laplace transform of the given funct...
 7.15: In 1117, determine the inverse Laplace transform of the given funct...
 7.16: In 1117, determine the inverse Laplace transform of the given funct...
 7.17: In 1117, determine the inverse Laplace transform of the given funct...
 7.18: Find the Taylor series for about t 0. Then, assuming that the Lapla...
 7.19: In 1924, solve the given initial value problem for using the method...
 7.20: In 1924, solve the given initial value problem for using the method...
 7.21: In 1924, solve the given initial value problem for using the method...
 7.22: In 1924, solve the given initial value problem for using the method...
 7.23: In 1924, solve the given initial value problem for using the method...
 7.24: In 1924, solve the given initial value problem for using the method...
 7.25: In 25 and 26, find solutions to the given initial value problem.
 7.26: In 25 and 26, find solutions to the given initial value problem.
 7.27: In 27 and 28, solve the given equation for y(t).
 7.28: In 27 and 28, solve the given equation for y(t).
 7.29: A linear system is governed by Find the transfer function and the i...
 7.30: Solve the symbolic initial value problem
 7.31: In 31 and 32, use Laplace transforms to solve the given system
 7.32: In 31 and 32, use Laplace transforms to solve the given system
Solutions for Chapter 7: Fundamentals of Differential Equations and Boundary Value Problems 6th Edition
Full solutions for Fundamentals of Differential Equations and Boundary Value Problems  6th Edition
ISBN: 9780321747747
Solutions for Chapter 7
Get Full SolutionsThis textbook survival guide was created for the textbook: Fundamentals of Differential Equations and Boundary Value Problems, edition: 6. Since 32 problems in chapter 7 have been answered, more than 2521 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Fundamentals of Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780321747747. Chapter 7 includes 32 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

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.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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