 11.5.11.1.91: (Coefficients) Derive the fonnula for en from An and Bn
 11.5.11.1.92: (Spring constant) What would happen to the amplitudes en in Example...
 11.5.11.1.93: (Damping) In Example I change c to 0.02 and discuss how this change...
 11.5.11.1.94: (Input) What would happen in Example I if we replaced ret) with its...
 11.5.11.1.95: Find a general solution of the ODE y" + w2y = ret) with r(t) as giv...
 11.5.11.1.96: Find a general solution of the ODE y" + w2y = ret) with r(t) as giv...
 11.5.11.1.97: Find a general solution of the ODE y" + w2y = ret) with r(t) as giv...
 11.5.11.1.98: Find a general solution of the ODE y" + w2y = ret) with r(t) as giv...
 11.5.11.1.99: Find a general solution of the ODE y" + w2y = ret) with r(t) as giv...
 11.5.11.1.100: Find a general solution of the ODE y" + w2y = ret) with r(t) as giv...
 11.5.11.1.101: Find a general solution of the ODE y" + w2y = ret) with r(t) as giv...
 11.5.11.1.102: (CAS Program) Write a program for solving the ODE just considered a...
 11.5.11.1.103: (Sign of coefficients) Some An in Example 1 are positive and some n...
 11.5.11.1.104: Find the steadystate oscillation of y" + c/ + Y = r(t) with c > 0 ...
 11.5.11.1.105: Find the steadystate oscillation of y" + c/ + Y = r(t) with c > 0 ...
 11.5.11.1.106: Find the steadystate oscillation of y" + c/ + Y = r(t) with c > 0 ...
 11.5.11.1.107: Find the steadystate oscillation of y" + c/ + Y = r(t) with c > 0 ...
 11.5.11.1.108: CAS EXPERIMENT. Maximum Output Term. Graph and discus~ outputs of y...
 11.5.11.1.109: Find the steadystate current I(t) in the RLCcircuit in Fig. 272, ...
 11.5.11.1.110: Find the steadystate current I(t) in the RLCcircuit in Fig. 272, ...
Solutions for Chapter 11.5: Forced Oscillations
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 11.5: Forced Oscillations
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Since 20 problems in chapter 11.5: Forced Oscillations have been answered, more than 49005 students have viewed full stepbystep solutions from this chapter. Chapter 11.5: Forced Oscillations includes 20 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Cholesky factorization
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).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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).

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.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).