- 3.8.1: In each of 1 through 3, write the given expression as a product of ...
- 3.8.2: In each of 1 through 3, write the given expression as a product of ...
- 3.8.3: In each of 1 through 3, write the given expression as a product of ...
- 3.8.4: A mass of 5 kg stretches a spring 10 cm. The mass is acted on by an...
- 3.8.5: a. Find the solution of the initial value problem in 4. b. Identify...
- 3.8.6: A mass that weighs 8 lb stretches a spring 6 in. The system is acte...
- 3.8.7: A spring is stretched 6 in by a mass that weighs 8 lb. The mass is ...
- 3.8.8: A spring-mass system has a spring constant of 3 N/m. A mass of 2 kg...
- 3.8.9: In this problem we ask you to supply some of the details in the ana...
- 3.8.10: Find the velocity of the steady-state response given by equation (1...
- 3.8.11: Find the solution of the initial value problem u__ + u = F(t), u(0)...
- 3.8.12: A series circuit has a capacitor of 0.25 106 F, a resistor of 5103 ...
- 3.8.13: Consider the forced but undamped system described by the initial va...
- 3.8.14: Consider the vibrating system described by the initial value proble...
- 3.8.15: For the initial value problem in 13, plot u_ versus u for = 0.7, = ...
- 3.8.16: In each of these problems: G a. Plot the given forcing function F(t...
- 3.8.17: In each of these problems: G a. Plot the given forcing function F(t...
- 3.8.18: In each of these problems: G a. Plot the given forcing function F(t...
- 3.8.19: A spring-mass system with a hardening spring ( of Section 3.7) is a...
Solutions for Chapter 3.8: Forced Periodic Vibrations
Full solutions for Elementary Differential Equations and Boundary Value Problems | 11th Edition
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.
peA) = det(A - AI) has peA) = zero matrix.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
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.
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 S-I AS = A = eigenvalue matrix.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
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 - ).
Outer product uv T
= column times row = rank one matrix.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.