 1.8.1: A given substance satisfies the exponential growth law (1). Show th...
 1.8.2: A substance x multiplies exponentially, and a given quantity of the...
 1.8.3: A substance x decays exponentially, and only half of the given quan...
 1.8.4: The equation p'= apa, a > 1, is proposed as a model of the populati...
 1.8.5: A cancerous tumor satisfies the Gompertzian relation (3). Originall...
 1.8.6: A tracer dose of radioactive iodine 1311 is injected into the blood...
 1.8.7: Industrial waste is pumped into a tank containing 1000 gallons of w...
 1.8.8: A tank contains 300 gallons of water and 100 gallons of pollutants....
 1.8.9: Consider a tank containing, at time t =0, Qo lb of salt dissolved i...
 1.8.10: A room containing 1000 cubic feet of air is originally free of carb...
 1.8.11: A 500 gallon tank originally contains 100 gallons of fresh water. B...
 1.8.12: In Exercises 1217, find the orthogonal trajectories of the given f...
 1.8.13: In Exercises 1217, find the orthogonal trajectories of the given f...
 1.8.14: In Exercises 1217, find the orthogonal trajectories of the given f...
 1.8.15: In Exercises 1217, find the orthogonal trajectories of the given f...
 1.8.16: In Exercises 1217, find the orthogonal trajectories of the given f...
 1.8.17: In Exercises 1217, find the orthogonal trajectories of the given f...
 1.8.18: The presence of toxins in a certain medium destroys a strain of bac...
 1.8.19: Many savings banks now advertise continuous compounding of interest...
Solutions for Chapter 1.8: The dynamics of tumor growth, mixing problems and orthogonal trajectories
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 1.8: The dynamics of tumor growth, mixing problems and orthogonal trajectories
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Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

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

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].

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

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).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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 •

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).

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.