- 5.1.1: Consider the three systems (i) dx dt = 2x + y dy dt = y + x2 (ii) d...
- 5.1.2: Consider the following three systems: (i) dx dt = 3 sin x + y dy dt...
- 5.1.3: Consider the system dx dt = 2x + y dy dt = y + x2. (a) Find the lin...
- 5.1.4: Consider the system dx dt = x dy dt = 4x3 + y. (a) Show that the or...
- 5.1.5: Consider the system in Exercise 4. (a) Find the general solution of...
- 5.1.6: For the competing species population model dx dt = 2x 1 x 2 x y dy ...
- 5.1.7: dx dt = x(x 3y + 150) dy dt = y(2x y + 100)
- 5.1.8: dx dt = x(10 x y) dy dt = y(30 2x y)
- 5.1.9: dx dt = x(100 x 2y) dy dt = y(150 x 6y)
- 5.1.10: dx dt = x(x y + 100) dy dt = y(x2 y2 + 2500)
- 5.1.11: dx dt = x(x y + 40) dy dt = y(x2 y2 + 2500)
- 5.1.12: dx dt = x(4x y + 160) dy dt = y(x2 y2 + 2500)
- 5.1.13: dx dt = x(8x 6y + 480) dy dt = y(x2 y2 + 2500)
- 5.1.14: dx dt = x(2 x y) dy dt = y(y x2)
- 5.1.15: dx dt = x(2 x y) dy dt = y(y x)
- 5.1.16: dx dt = x(x 1) dy dt = y(x2 y)
- 5.1.17: Consider the system dx dt = x3 dy dt = y + y2. It has equilibrium p...
- 5.1.18: If a nonlinear system depends on a parameter, then the equilibrium ...
- 5.1.19: Continuing the study of the nonlinear system given in Exercise 18,(...
- 5.1.20: dxdt = y x2dydt = y a
- 5.1.21: dx dt = y x2 dy dt = a
- 5.1.22: dx dt = y x2 dy dt = y x a
- 5.1.23: dx dt = y ax3 dy dt = y x
- 5.1.24: dx dt = y x2 + a dy dt = y + x2 a
- 5.1.25: dx dt = y x2 + a dy dt = y + x2
- 5.1.26: The system dx dt = x(x y + 70) dy dt = y(2x y + a) is a model for a...
- 5.1.27: Suppose two species X and Y are to be introduced to an island. It i...
- 5.1.28: For the two species X and Y of Exercise 27, suppose that both X and...
- 5.1.29: For the species X and Y in Exercises 27 and 28, suppose that X repr...
- 5.1.30: Suppose two similar countries Y and Z are engaged in an arms race. ...
Solutions for Chapter 5.1: EQUILIBRIUM POINT ANALYSIS
Full solutions for Differential Equations 00 | 4th Edition
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.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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].
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
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.
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].
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.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. 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.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Every v in V is orthogonal to every w in W.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.