 5.2.5.1.59: (a) The beam is embedded at its left end and free at its right end,...
 5.2.5.1.60: (a) The beam is simply supported at both ends, and w(x) w0, 0 x L. ...
 5.2.5.1.61: (a) The beam is embedded at its left end and simply supported at it...
 5.2.5.1.62: (a) The beam is embedded at its left end and simply supported at it...
 5.2.5.1.63: (a) The beam is simply supported at both ends, and w(x) w0 x, 0 x L...
 5.2.5.1.64: (a) Find the maximum deflection of the cantilever beam in 1. (b) Ho...
 5.2.5.1.65: A cantilever beam of length L is embedded at its right end, and a h...
 5.2.5.1.66: When a compressive instead of a tensile force is applied at the fre...
 5.2.5.1.67: In 918 find the eigenvalues and eigenfunctions for the given bounda...
 5.2.5.1.68: In 918 find the eigenvalues and eigenfunctions for the given bounda...
 5.2.5.1.69: In 918 find the eigenvalues and eigenfunctions for the given bounda...
 5.2.5.1.70: In 918 find the eigenvalues and eigenfunctions for the given bounda...
 5.2.5.1.71: In 918 find the eigenvalues and eigenfunctions for the given bounda...
 5.2.5.1.72: In 918 find the eigenvalues and eigenfunctions for the given bounda...
 5.2.5.1.73: In 918 find the eigenvalues and eigenfunctions for the given bounda...
 5.2.5.1.74: In 918 find the eigenvalues and eigenfunctions for the given bounda...
 5.2.5.1.75: In 918 find the eigenvalues and eigenfunctions for the given bounda...
 5.2.5.1.76: In 918 find the eigenvalues and eigenfunctions for the given bounda...
 5.2.5.1.77: In 19 and 20 find the eigenvalues and eigenfunctions for the given ...
 5.2.5.1.78: In 19 and 20 find the eigenvalues and eigenfunctions for the given ...
 5.2.5.1.79: Consider Figure 5.2.6. Where should physical restraints be placed o...
 5.2.5.1.80: The critical loads of thin columns depend on the end conditions of ...
 5.2.5.1.81: As was mentioned in 22, the differential equation (5) that governs ...
 5.2.5.1.82: Suppose that a uniform thin elastic column is hinged at the end x 0...
 5.2.5.1.83: Consider the boundaryvalue problem introduced in the construction ...
 5.2.5.1.84: When the magnitude of tension T is not constant, then a model for t...
 5.2.5.1.85: Temperature in a Sphere Consider two concentric spheres of radius r...
 5.2.5.1.86: Temperature in a Ring The temperature u(r) in the circular ring sho...
 5.2.5.1.87: Simple Harmonic Motion The model mx kx 0 for simple harmonic motion...
 5.2.5.1.88: Damped Motion Assume that the model for the spring/mass system in i...
 5.2.5.1.89: In 31 and 32 determine whether it is possible to find values y0 and...
 5.2.5.1.90: In 31 and 32 determine whether it is possible to find values y0 and...
 5.2.5.1.91: Consider the boundaryvalue problem (a) The type of boundary condit...
 5.2.5.1.92: Show that the eigenvalues and eigenfunctions of the boundaryvalue ...
 5.2.5.1.93: Use a CAS to plot graphs to convince yourself that the equation tan...
 5.2.5.1.94: Use a rootfinding application of a CAS to approximate the first fo...
 5.2.5.1.95: In 37 and 38 find the eigenvalues and eigenfunctions of the given b...
 5.2.5.1.96: In 37 and 38 find the eigenvalues and eigenfunctions of the given b...
Solutions for Chapter 5.2: Modeling with HigherOrder Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 5.2: Modeling with HigherOrder Differential Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. This expansive textbook survival guide covers the following chapters and their solutions. Since 38 problems in chapter 5.2: Modeling with HigherOrder Differential Equations have been answered, more than 21002 students have viewed full stepbystep solutions from this chapter. Chapter 5.2: Modeling with HigherOrder Differential Equations includes 38 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Graph G.
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.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Iterative method.
A sequence of steps intended to approach the desired solution.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

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

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.