 10.4.1: 16 Each of the nonlinear systems has an equilibrium at sx^1, x^2d s...
 10.4.2: 16 Each of the nonlinear systems has an equilibrium at sx^1, x^2d s...
 10.4.3: 16 Each of the nonlinear systems has an equilibrium at sx^1, x^2d s...
 10.4.4: 16 Each of the nonlinear systems has an equilibrium at sx^1, x^2d s...
 10.4.5: 16 Each of the nonlinear systems has an equilibrium at sx^1, x^2d s...
 10.4.6: 16 Each of the nonlinear systems has an equilibrium at sx^1, x^2d s...
 10.4.7: 712 Find all equilibria. Then find the linearization near each equi...
 10.4.8: 712 Find all equilibria. Then find the linearization near each equi...
 10.4.9: 712 Find all equilibria. Then find the linearization near each equi...
 10.4.10: 712 Find all equilibria. Then find the linearization near each equi...
 10.4.11: 712 Find all equilibria. Then find the linearization near each equi...
 10.4.12: 712 Find all equilibria. Then find the linearization near each equi...
 10.4.13: 1318 A Jacobian matrix and two equlibria are given. Determine if ea...
 10.4.14: 1318 A Jacobian matrix and two equlibria are given. Determine if ea...
 10.4.15: 1318 A Jacobian matrix and two equlibria are given. Determine if ea...
 10.4.16: 1318 A Jacobian matrix and two equlibria are given. Determine if ea...
 10.4.17: 1318 A Jacobian matrix and two equlibria are given. Determine if ea...
 10.4.18: 1318 A Jacobian matrix and two equlibria are given. Determine if ea...
 10.4.19: 1923 Find all equilibria and determine their local stability proper...
 10.4.20: 1923 Find all equilibria and determine their local stability proper...
 10.4.21: 1923 Find all equilibria and determine their local stability proper...
 10.4.22: 1923 Find all equilibria and determine their local stability proper...
 10.4.23: 1923 Find all equilibria and determine their local stability proper...
 10.4.24: 2425 Find all equilibria and determine their stability properties. ...
 10.4.25: 2425 Find all equilibria and determine their stability properties. ...
 10.4.26: Cell cycle In Exercise 7.Review.25 a model for the cell cycle was i...
 10.4.27: Competitioncolonization models In Exercise 7.Review.23 a metapopul...
 10.4.28: Gene regulation The model of gene regulation from Section 10.3 is o...
 10.4.29: 2931 Consumer resource models often have the following general form...
 10.4.30: 2931 Consumer resource models often have the following general form...
 10.4.31: 2931 Consumer resource models often have the following general form...
 10.4.32: The KermackMcKendrick equations describe the outbreak of an infect...
 10.4.33: The MichaelisMenten equations describe a biochemical reaction in w...
 10.4.34: Stability of Caribbean reefs Coral and macroalgae compete for space...
 10.4.35: FitzhughNagumo equations Consider the following alternative form o...
Solutions for Chapter 10.4: Systems of Nonlinear Differential Equations
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 10.4: Systems of Nonlinear Differential Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. Since 35 problems in chapter 10.4: Systems of Nonlinear Differential Equations have been answered, more than 27424 students have viewed full stepbystep solutions from this chapter. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. Chapter 10.4: Systems of Nonlinear Differential Equations includes 35 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Outer product uv T
= column times row = rank one matrix.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.