 5.5.1.123: If a mass weighing 10 pounds stretches a spring 2.5 feet, a mass we...
 5.5.1.124: The period of simple harmonic motion of mass weighing 8 pounds atta...
 5.5.1.125: The differential equation of a spring/mass system is x 16x 0. If th...
 5.5.1.126: Pure resonance cannot take place in the presence of a damping force.
 5.5.1.127: In the presence of a damping force, the displacements of a mass on ...
 5.5.1.128: A mass on a spring whose motion is critically damped can possibly p...
 5.5.1.129: At critical damping any increase in damping will result in an system.
 5.5.1.130: If simple harmonic motion is described by , the phase angle f is wh...
 5.5.1.131: A solution of the BVP when l 8 is y because .
 5.5.1.132: A solution of the BVP when l 36 is y because .
 5.5.1.133: A free undamped spring/mass system oscillates with a period of 3 se...
 5.5.1.134: A mass weighing 12 pounds stretches a spring 2 feet. The mass is in...
 5.5.1.135: A force of 2 pounds stretches a spring 1 foot. With one end held fi...
 5.5.1.136: A mass weighing 32 pounds stretches a spring 6 inches. The mass mov...
 5.5.1.137: A spring with constant k 2 is suspended in a liquid that offers a d...
 5.5.1.138: The vertical motion of a mass attached to a spring is described by ...
 5.5.1.139: A mass weighing 4 pounds stretches a spring 18 inches. A periodic f...
 5.5.1.140: Find a particular solution for x 2lx v2x A, where A is a constant f...
 5.5.1.141: A mass weighing 4 pounds is suspended from a spring whose constant ...
 5.5.1.142: (a) Two springs are attached in series as shown in Figure 5.R.1. If...
 5.5.1.143: A series circuit contains an inductance of L 1 h, a capacitance of ...
 5.5.1.144: (a) Show that the current i(t) in an LRCseries circuit satisfies ,...
 5.5.1.145: Consider the boundaryvalue problem . Show that except for the case...
 5.5.1.146: A bead is constrained to slide along a frictionless rod of length L...
 5.5.1.147: Suppose a mass m lying on a flat dry frictionless surface is attach...
 5.5.1.148: Suppose the mass m on the flat, dry, frictionless surface in is att...
 5.5.1.149: Suppose the mass m in the spring/mass system in slides over a dry s...
 5.5.1.150: For simplicity, let us assume in that and (a) Find the displacement...
Solutions for Chapter 5: Modeling with HigherOrder Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 5: Modeling with HigherOrder Differential Equations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Since 28 problems in chapter 5: Modeling with HigherOrder Differential Equations have been answered, more than 20275 students have viewed full stepbystep solutions from this chapter. Chapter 5: Modeling with HigherOrder Differential Equations includes 28 full stepbystep solutions. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (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.

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.