 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 LotkaVolterra equations to model populations of sharks S...
 7.9: What is a nullcline?
 7.10: (a) Write LotkaVolterra competition equations for two competing fi...
 7.11: 1112 Solve the initialvalue problem. 11. dr dt 1 2tr r, rs0d 5 12....
 7.12: 1112 Solve the initialvalue 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 BrentanoStevens 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: Predatorprey 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: Competitioncolonization 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
ISBN: 9781133109631
Solutions for Chapter 7: 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. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. Since 25 problems in chapter 7: Differential Equations have been answered, more than 25064 students have viewed full stepbystep solutions from this chapter. Chapter 7: Differential Equations includes 25 full stepbystep solutions.

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.

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. Blockdiagonal: 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.

GramSchmidt 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, offdiagonal entries = 0.

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Partial pivoting.
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 = Q1 = Q.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Schwarz inequality
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.