 2.8.2.1.153: Find the steadystate oscillation of the massspring system modeled...
 2.8.2.1.154: Find the steadystate oscillation of the massspring system modeled...
 2.8.2.1.155: Find the steadystate oscillation of the massspring system modeled...
 2.8.2.1.156: Find the steadystate oscillation of the massspring system modeled...
 2.8.2.1.157: Find the steadystate oscillation of the massspring system modeled...
 2.8.2.1.158: Find the steadystate oscillation of the massspring system modeled...
 2.8.2.1.159: Find the steadystate oscillation of the massspring system modeled...
 2.8.2.1.160: Find the steadystate oscillation of the massspring system modeled...
 2.8.2.1.161: Find the transient motion of the massspring system modeled by the ...
 2.8.2.1.162: Find the transient motion of the massspring system modeled by the ...
 2.8.2.1.163: Find the transient motion of the massspring system modeled by the ...
 2.8.2.1.164: Find the transient motion of the massspring system modeled by the ...
 2.8.2.1.165: Find the transient motion of the massspring system modeled by the ...
 2.8.2.1.166: Find the transient motion of the massspring system modeled by the ...
 2.8.2.1.167: Find the motion of the massspring system modeled by the ODE and in...
 2.8.2.1.168: Find the motion of the massspring system modeled by the ODE and in...
 2.8.2.1.169: Find the motion of the massspring system modeled by the ODE and in...
 2.8.2.1.170: Find the motion of the massspring system modeled by the ODE and in...
 2.8.2.1.171: Find the motion of the massspring system modeled by the ODE and in...
 2.8.2.1.172: Find the motion of the massspring system modeled by the ODE and in...
 2.8.2.1.173: (Beats) Derive the formula after (12) from (12). Can there be beats...
 2.8.2.1.174: (Beats) How does the graph ofthe solution in Prob. 20 change if you...
 2.8.2.1.175: WRITING PROJECT. Free and Forced Vibrations. Write a condensed repo...
 2.8.2.1.176: CAS EXPERIMENT. Undamped Vibrations. (a) Solve the initial value pr...
 2.8.2.1.177: TEAM PROJECT. Practical Resonance. (a) Give a detailed derivation o...
 2.8.2.1.178: (Gun barrel) Solve { I y " + Y = ifO~t~7T o ift>7T, .1'(0) = y' (0)...
Solutions for Chapter 2.8: Modeling: Forced Oscillations. Resonance
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 2.8: Modeling: Forced Oscillations. Resonance
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Since 26 problems in chapter 2.8: Modeling: Forced Oscillations. Resonance have been answered, more than 45669 students have viewed full stepbystep solutions from this chapter. Chapter 2.8: Modeling: Forced Oscillations. Resonance includes 26 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions.

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.)

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

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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 .

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Outer product uv T
= column times row = rank one matrix.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.