- 3.8.1: In each of 1 through 4 write the given expression as a product of t...
- 3.8.2: In each of 1 through 4 write the given expression as a product of t...
- 3.8.3: In each of 1 through 4 write the given expression as a product of t...
- 3.8.4: In each of 1 through 4 write the given expression as a product of t...
- 3.8.5: A mass weighing 4 lb stretches a spring 1.5 in. The mass is displac...
- 3.8.6: A mass of 5 kg stretches a spring 10 cm. The mass is acted on by an...
- 3.8.7: (a) Find the solution of 5. (b) Plot the graph of the solution. (c)...
- 3.8.8: (a) Find the solution of the initial value problem in 6. (b) Identi...
- 3.8.9: If an undamped springmass system with a mass that weighs 6 lb and a...
- 3.8.10: A mass that weighs 8 lb stretches a spring 6 in. The system is acte...
- 3.8.11: A spring is stretched 6 in by a mass that weighs 8 lb. The mass is ...
- 3.8.12: A springmass system has a spring constant of 3 N/m. A mass of 2 kg ...
- 3.8.13: In this problem we ask you to supply some of the details in the ana...
- 3.8.14: Find the velocity of the steady state response given by Eq. (10). T...
- 3.8.15: Find the solution of the initial value problem u + u = F(t), u(0) =...
- 3.8.16: A series circuit has a capacitor of 0.25 106 F, a resistor of 5 103...
- 3.8.17: Consider a vibrating system described by the initial value problem ...
- 3.8.18: Consider the forced but undamped system described by the initial va...
- 3.8.19: Consider the vibrating system described by the initial value proble...
- 3.8.20: For the initial value problem in plot u versus u for = 0.7, = 0.8, ...
- 3.8.21: 21 through 23 deal with the initial value problem u + 0.125u + 4u =...
- 3.8.22: 21 through 23 deal with the initial value problem u + 0.125u + 4u =...
- 3.8.23: 21 through 23 deal with the initial value problem u + 0.125u + 4u =...
- 3.8.24: A springmass system with a hardening spring ( of Section 3.7) is ac...
- 3.8.25: Suppose that the system of is modified to include a damping term an...
Solutions for Chapter 3.8: Forced Vibrations
Full solutions for Elementary Differential Equations and Boundary Value Problems | 9th Edition
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
A = CTC = (L.J]))(L.J]))T for positive definite A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
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.
A symmetric matrix with eigenvalues of both signs (+ and - ).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Outer product uv T
= column times row = rank one matrix.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.