- 4.10.15E: In 13–16 proceed as in Example 3 and obtain the first six nonzero t...
- 4.10.16E: In 13–16 proceed as in Example 3 and obtain the first six nonzero t...
- 4.10.1E: In 1 and 2 verify that y1 and y2 are solutions of the given differe...
- 4.10.2E: In 1 and 2 verify that y1 and y2 are solutions of the given differe...
- 4.10.3E: In 3–8 solve the given differential equation by using the substitution
- 4.10.4E: In 3–8 solve the given differential equation by using the substitution
- 4.10.5E: In 3–8 solve the given differential equation by using the substitution
- 4.10.6E: In 3–8 solve the given differential equation by using the substitution
- 4.10.7E: In 3–8 solve the given differential equation by using the substitution
- 4.10.8E: In 3–8 solve the given differential equation by using the substitution
- 4.10.9E: In the given initial-value problem.
- 4.10.10E: In the given initial-value problem.
- 4.10.11E: Consider the initial-value problem (a) Use the DE and a numerical s...
- 4.10.12E: Find two solutions of the initial-value problem Use a numerical sol...
- 4.10.13E: In 11 and 12 show that the substitution leads to a Bernoulli equati...
- 4.10.14E: In 11 and 12 show that the substitution leads to a Bernoulli equati...
- 4.10.17E: In 13–16 proceed as in Example 3 and obtain the first six nonzero t...
- 4.10.18E: In 13–16 proceed as in Example 3 and obtain the first six nonzero t...
- 4.10.19E: In calculus the curvature of a curve that is defined by a function ...
- 4.10.20E: In we saw that cos x and ex were solutions of the nonlinear equatio...
- 4.10.21E: Discuss how the method of reduction of order considered in this sec...
- 4.10.22E: Discuss how to find an alternative two-parameter family of solution...
- 4.10.23E: Motion in a Force Field A mathematical model for the position x(t) ...
- 4.10.24E: A mathematical model for the position x(t) of a moving object is Us...
Solutions for Chapter 4.10: A First Course in Differential Equations with Modeling Applications 10th Edition
Full solutions for A First Course in Differential Equations with Modeling Applications | 10th Edition
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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.
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.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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