 1.3.1.1.112: Under the same assumptions that underlie the model in (1), determin...
 1.3.1.1.113: The population model given in (1) fails to take death into consider...
 1.3.1.1.114: Using the concept of net rate introduced in 2, determine a model fo...
 1.3.1.1.115: Modify the model in for net rate at which the population P(t) of a ...
 1.3.1.1.116: A cup of coffee cools according to Newtons law of cooling (3). Use ...
 1.3.1.1.117: The ambient temperature Tm in (3) could be a function of time t. Su...
 1.3.1.1.118: Suppose a student carrying a fl virus returns to an isolated colleg...
 1.3.1.1.119: At a time denoted as t 0 a technological innovation is introduced i...
 1.3.1.1.120: Suppose that a large mixing tank initially holds 300 gallons of wat...
 1.3.1.1.121: Suppose that a large mixing tank initially holds 300 gallons of wat...
 1.3.1.1.122: What is the differential equation in 10, if the wellstirred soluti...
 1.3.1.1.123: Generalize the model given in equation (8) on page 24 by assuming t...
 1.3.1.1.124: Suppose water is leaking from a tank through a circular hole of are...
 1.3.1.1.125: The rightcircular conical tank shown in Figure 1.3.13 loses water ...
 1.3.1.1.126: A series circuit contains a resistor and an inductor as shown in Fi...
 1.3.1.1.127: A series circuit contains a resistor and a capacitor as shown in Fi...
 1.3.1.1.128: For highspeed motion through the airsuch as the skydiver shown in ...
 1.3.1.1.129: A cylindrical barrel s feet in diameter of weight w lb is floating ...
 1.3.1.1.130: After a mass m is attached to a spring, it stretches it s units and...
 1.3.1.1.131: In 19, what is a differential equation for the displacement x(t) if...
 1.3.1.1.132: A small singlestage rocket is launched vertically as shown in Figu...
 1.3.1.1.133: In 21, the mass m(t) is the sum of three different masses: where mp...
 1.3.1.1.134: By Newtons universal law of gravitation the freefall acceleration ...
 1.3.1.1.135: Suppose a hole is drilled through the center of the Earth and a bow...
 1.3.1.1.136: Learning Theory In the theory of learning, the rate at which a subj...
 1.3.1.1.137: Forgetfulness In assume that the rate at which material is forgotte...
 1.3.1.1.138: Infusion of a Drug A drug is infused into a patients bloodstream at...
 1.3.1.1.139: Tractrix A person P, starting at the origin, moves in the direction...
 1.3.1.1.140: Reflecting Surface Assume that when the plane curve C shown in Figu...
 1.3.1.1.141: Reread in Exercises 1.1 and then give an explicit solution P(t) for...
 1.3.1.1.142: Reread the sentence following equation (3) and assume that Tm is a ...
 1.3.1.1.143: Reread the discussion leading up to equation (8). If we assume that...
 1.3.1.1.144: Population Model The differential equation where k is a positive co...
 1.3.1.1.145: Rotating Fluid As shown in Figure 1.3.24(a), a rightcircular cylind...
 1.3.1.1.146: Falling Body In 23, suppose r R s, where s is the distance from the...
 1.3.1.1.147: Raindrops Keep Falling In meteorology the term virga refers to fall...
 1.3.1.1.148: Let It Snow The snowplow problem is a classic and appears in many d...
 1.3.1.1.149: Reread this section and classify each mathematical model as linear ...
Solutions for Chapter 1.3: INTRODUCTION TO DIFFERENTIAL EQUATIONS
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 1.3: INTRODUCTION TO DIFFERENTIAL EQUATIONS
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 38 problems in chapter 1.3: INTRODUCTION TO DIFFERENTIAL EQUATIONS have been answered, more than 23121 students have viewed full stepbystep solutions from this chapter. Chapter 1.3: INTRODUCTION TO DIFFERENTIAL EQUATIONS includes 38 full stepbystep solutions. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

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).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

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

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.