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# Solutions for Chapter 5.2: Modeling with Higher-Order Differential Equations

## Full solutions for Differential Equations with Boundary-Value Problems, | 8th Edition

ISBN: 9781111827069

Solutions for Chapter 5.2: Modeling with Higher-Order Differential Equations

Solutions for Chapter 5.2
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##### ISBN: 9781111827069

This textbook survival guide was created for the textbook: Differential Equations with Boundary-Value Problems,, edition: 8. Differential Equations with Boundary-Value 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 Higher-Order Differential Equations have been answered, more than 21002 students have viewed full step-by-step solutions from this chapter. Chapter 5.2: Modeling with Higher-Order Differential Equations includes 38 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• 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 = Fn-l + 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.

• Gram-Schmidt 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.

• lA-II = 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+ (Moore-Penrose 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.

• Skew-symmetric 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. A-I 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.

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