 10.5.1: For 111, determine f (ta)for the given function f and the given con...
 10.5.2: For 111, determine f (ta)for the given function f and the given con...
 10.5.3: For 111, determine f (ta)for the given function f and the given con...
 10.5.4: For 111, determine f (ta)for the given function f and the given con...
 10.5.5: For 111, determine f (ta)for the given function f and the given con...
 10.5.6: For 111, determine f (ta)for the given function f and the given con...
 10.5.7: For 111, determine f (ta)for the given function f and the given con...
 10.5.8: For 111, determine f (ta)for the given function f and the given con...
 10.5.9: For 111, determine f (ta)for the given function f and the given con...
 10.5.10: For 111, determine f (ta)for the given function f and the given con...
 10.5.11: For 111, determine f (ta)for the given function f and the given con...
 10.5.12: For 1217, determine f (t).f (t 1) = (t 1)2.
 10.5.13: For 1217, determine f (t).f (t 1) = (t 2)2.
 10.5.14: For 1217, determine f (t).f (t 2) = (t 2)e3(t2).
 10.5.15: For 1217, determine f (t).f (t 1) = t sin[3(t 1)].
 10.5.16: For 1217, determine f (t).f (t 3) = te(t3).
 10.5.17: For 1217, determine f (t).f (t 4) = t + 1(t 1)2 + 4.
 10.5.18: For 1827, determine the Laplace transform of ff (t) = e3t cos 4t.
 10.5.19: For 1827, determine the Laplace transform of f
 10.5.20: For 1827, determine the Laplace transform of ff (t) = te2t.
 10.5.21: For 1827, determine the Laplace transform of ff (t) = 3tet.
 10.5.22: For 1827, determine the Laplace transform of ff (t) = t3e4t.
 10.5.23: For 1827, determine the Laplace transform of ff (t) = et te2t.
 10.5.24: For 1827, determine the Laplace transform of ff (t) = 2e3t sin t + ...
 10.5.25: For 1827, determine the Laplace transform of ff (t) = e2t(1 sin2 t).
 10.5.26: For 1827, determine the Laplace transform of ff (t) = t2(et 3).
 10.5.27: For 1827, determine the Laplace transform of ff (t) = t2(et 3).
 10.5.28: For 2842, determine L1[F].F(s) = 1(s 3)2 .
 10.5.29: For 2842, determine L1[F].
 10.5.30: For 2842, determine L1[F].F(s) = 2s + 3.
 10.5.31: For 2842, determine L1[F].
 10.5.32: For 2842, determine L1[F].F(s) = s + 2(s + 2)2 + 9.
 10.5.33: For 2842, determine L1[F].F(s) = s(s 3)2 + 4.
 10.5.34: For 2842, determine L1[F].
 10.5.35: For 2842, determine L1[F].F(s) = 6s2 + 2s + 2
 10.5.36: For 2842, determine L1[F].F(s) = s 2s2 + 2s + 26
 10.5.37: For 2842, determine L1[F].
 10.5.38: For 2842, determine L1[F].F(s) = s(s + 1)2 + 4
 10.5.39: For 2842, determine L1[F].F(s) = 2s + 3(s + 5)2 + 49.
 10.5.40: For 2842, determine L1[F].F(s) = 4s(s + 2)2 .
 10.5.41: For 2842, determine L1[F].F(s) = 2s + 1(s 1)2(s + 2).
 10.5.42: For 2842, determine L1[F].F(s) = 2s + 3s(s2 2s + 5).
 10.5.43: For 4353, solve the given initialvalue problem.y y = 8et, y(0) = 0...
 10.5.44: For 4353, solve the given initialvalue problem.
 10.5.45: For 4353, solve the given initialvalue problem.y y 2y = 6et, y(0) ...
 10.5.46: For 4353, solve the given initialvalue problem.y + y 2y = 3e2t, y(...
 10.5.47: For 4353, solve the given initialvalue problem.
 10.5.48: For 4353, solve the given initialvalue problem.y + 2y + y = 2et, y...
 10.5.49: For 4353, solve the given initialvalue problem.y 4y = 2tet, y(0) =...
 10.5.50: For 4353, solve the given initialvalue problem.y + 3y + 2y = 12te2...
 10.5.51: For 4353, solve the given initialvalue problem.
 10.5.52: For 4353, solve the given initialvalue problem.y y = 8et sin 2t, y...
 10.5.53: For 4353, solve the given initialvalue problem.y +2y 3y = 26e2t co...
 10.5.54: Solve the initialvalue problem x 1 = 2x1 x2, x 2 = x1 + 2x2, x1(0)...
 10.5.55: Solve the initialvalue problem x 1 = 3x1 + 2x2, x 2 = x1 + 4x2, x1...
Solutions for Chapter 10.5: The First Shifting Theorem
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 10.5: The First Shifting Theorem
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, edition: 4. Chapter 10.5: The First Shifting Theorem includes 55 full stepbystep solutions. Since 55 problems in chapter 10.5: The First Shifting Theorem have been answered, more than 20004 students have viewed full stepbystep solutions from this chapter. Differential Equations was written by and is associated to the ISBN: 9780321964670.

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

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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Column space C (A) =
space of all combinations of the columns of A.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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

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.

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

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.

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

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 .

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

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.