 10.1: What is the difference between an autonomous and a nonautonomous sy...
 10.2: What is an equilibrium of a system of differential equations?
 10.3: Explain the difference between local and global stability in system...
 10.4: What is the difference between the solution of an initialvalue prob...
 10.5: What does the Existence and Uniqueness Theorem tell us about homoge...
 10.6: Explain the Superposition Principle.
 10.7: Explain the difference between nullclines and eigenvectors in syste...
 10.8: What is the linearization of a system of nonlinear differential equ...
 10.9: Explain what a Jacobian matrix is.
 10.10: What do the eigenvalues of a Jacobian matrix from a system of nonli...
 10.11: 912 Solve the initialvalue problem dxydt Ax with xs0d x0. A c1 10 ...
 10.12: 912 Solve the initialvalue problem dxydt Ax with xs0d x0. A c21 12...
 10.13: 1318 Twocompartment mixing problems are similar to the mixing prob...
 10.14: 1318 Twocompartment mixing problems are similar to the mixing prob...
 10.15: 1318 Twocompartment mixing problems are similar to the mixing prob...
 10.16: 1318 Twocompartment mixing problems are similar to the mixing prob...
 10.17: 1318 Twocompartment mixing problems are similar to the mixing prob...
 10.18: 1318 Twocompartment mixing problems are similar to the mixing prob...
 10.19: 1922 A Jacobian matrix and equilibrium are given. Is the equilibriu...
 10.20: 1922 A Jacobian matrix and equilibrium are given. Is the equilibriu...
 10.21: 1922 A Jacobian matrix and equilibrium are given. Is the equilibriu...
 10.22: 1922 A Jacobian matrix and equilibrium are given. Is the equilibriu...
 10.23: 2326 Find all equilibria and determine their local stability proper...
 10.24: 2326 Find all equilibria and determine their local stability proper...
 10.25: 2326 Find all equilibria and determine their local stability proper...
 10.26: 2326 Find all equilibria and determine their local stability proper...
 10.27: The LotkaVolterra competition equations are dN1 dt S1 2 N1 1 N2 K1...
 10.28: The RosenzweigMacArthur model is a consumerresource model. A simpl...
 10.29: Habitat destruction The model of Exercise 10.4.27 can be extended t...
 10.30: Gene regulation The model of gene regulation from Section 10.3 is o...
 10.31: Sterile insect technique Sterile insects are sometimes released as ...
Solutions for Chapter 10: Systems of Linear Differential Equations
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 10: Systems of Linear Differential Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Since 31 problems in chapter 10: Systems of Linear Differential Equations have been answered, more than 25229 students have viewed full stepbystep solutions from this chapter. Chapter 10: Systems of Linear Differential Equations includes 31 full stepbystep solutions. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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].

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!).

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Lucas numbers
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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!