 3.7.1: In each of 1 through 4, determine 0, R, and so as to write the give...
 3.7.2: In each of 1 through 4, determine 0, R, and so as to write the give...
 3.7.3: In each of 1 through 4, determine 0, R, and so as to write the give...
 3.7.4: In each of 1 through 4, determine 0, R, and so as to write the give...
 3.7.5: A mass weighing 2 lb stretches a spring 6 in. If the mass is pulled...
 3.7.6: A mass of 100 g stretches a spring 5 cm. If the mass is set in moti...
 3.7.7: A mass weighing 3 lb stretches a spring 3 in. If the mass is pushed...
 3.7.8: A series circuit has a capacitor of 0.25 106 F and an inductor of 1...
 3.7.9: A mass of 20 g stretches a spring 5 cm. Suppose that the mass is al...
 3.7.10: . A mass weighing 16 lb stretches a spring 3 in. The mass is attach...
 3.7.11: A spring is stretched 10 cm by a force of 3 N. A mass of 2 kg is hu...
 3.7.12: A series circuit has a capacitor of 105 F, a resistor of 3 102 , an...
 3.7.13: A certain vibrating system satisfies the equation u + u + u = 0. Fi...
 3.7.14: Show that the period of motion of an undamped vibration of a mass h...
 3.7.15: Show that the solution of the initial value problemmu + u + ku = 0,...
 3.7.16: Show that A cos 0t + B sin 0t can be written in the form r sin(0t )...
 3.7.17: A mass weighing 8 lb stretches a spring 1.5 in. The mass is also at...
 3.7.18: If a series circuit has a capacitor of C = 0.8 106 F and an inducto...
 3.7.19: Assume that the system described by the equation mu + u + ku = 0 is...
 3.7.20: Assume that the system described by the equation mu + u + ku = 0 is...
 3.7.21: Logarithmic Decrement. (a) For the damped oscillation described by ...
 3.7.22: Referring to 21, find the logarithmic decrement of the system in 10.
 3.7.23: For the system in 17, suppose that = 3 and Td = 0.3 s. Referring to...
 3.7.24: The position of a certain springmass system satisfies the initial v...
 3.7.25: Consider the initial value problemu + u + u = 0, u(0) = 2, u(0) = 0...
 3.7.26: Consider the initial value problemmu + u + ku = 0, u(0) = u0, u(0) ...
 3.7.27: A cubic block of side l and mass density per unit volume is floatin...
 3.7.28: The position of a certain undamped springmass system satisfies the ...
 3.7.29: The position of a certain springmass system satisfies the initial v...
 3.7.30: In the absence of damping, the motion of a springmass system satisf...
 3.7.31: Suppose that a mass m slides without friction on a horizontal surfa...
 3.7.32: In the springmass system of 31, suppose that the spring force is no...
Solutions for Chapter 3.7: Mechanical and Electrical Vibrations
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 3.7: Mechanical and Electrical Vibrations
Get Full SolutionsChapter 3.7: Mechanical and Electrical Vibrations includes 32 full stepbystep solutions. Since 32 problems in chapter 3.7: Mechanical and Electrical Vibrations have been answered, more than 11399 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327. This expansive textbook survival guide covers the following chapters and their solutions.

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

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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.