- 7.1: a) What is a differential equation? (b) What is the order of a diff...
- 7.2: What can you say about the solutions of the equation y9 x 2 1 y 2 j...
- 7.3: What is a phase plot for the differential equation y9 ts yd?
- 7.4: What is a direction field for the differential equation y9 Fsx, yd?
- 7.5: Explain how Eulers method works
- 7.6: What is a separable differential equation? How do you solve it?
- 7.7: (a) Write the logistic equation. (b) Under what circumstances is th...
- 7.8: (a) Write Lotka-Volterra equations to model populations of sharks S...
- 7.9: What is a nullcline?
- 7.10: (a) Write Lotka-Volterra competition equations for two competing fi...
- 7.11: 1112 Solve the initial-value problem. 11. dr dt 1 2tr r, rs0d 5 12....
- 7.12: 1112 Solve the initial-value problem. 11. dr dt 1 2tr r, rs0d 5 12....
- 7.13: Seasonality and population dynamics The per capita growth rate of a...
- 7.14: easonality and population dynamics The per capita growth rate of a ...
- 7.15: Levins metapopulation model from Exercise 7.2.15 describes a popula...
- 7.16: The Brentano-Stevens Law in psychology models the way that a subjec...
- 7.17: Lung preoxygenation Some medical procedures require a patients airw...
- 7.18: A tank contains 100 L of pure water. Brine that contains 0.1 kg of ...
- 7.19: Hormone transport In lung physiology, the transport of a substance ...
- 7.20: Predator-prey dynamics Populations of birds and insects are modeled...
- 7.21: Suppose the model of Exercise 20 is replaced by the equations dx dt...
- 7.22: Cancer progression The development of many cancers, such as colorec...
- 7.23: Competition-colonization models The metapopulation model from Exerc...
- 7.24: Habitat destruction The model of Exercise 23 can be extended to inc...
- 7.25: Cell cycle dynamics The process of cell division is periodic, with ...
Solutions for Chapter 7: Differential Equations
Full solutions for Biocalculus: Calculus for Life Sciences | 1st Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
A = CTC = (L.J]))(L.J]))T for positive definite A.
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].
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
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.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
= Xl (column 1) + ... + xn(column n) = combination of columns.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
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.