 3.1.3.1.1: The population of a community is known to increase at a rate propor...
 3.1.3.1.2: Suppose it is known that the population of the community in is 10,0...
 3.1.3.1.3: The population of a town grows at a rate proportional to the popula...
 3.1.3.1.4: The population of bacteria in a culture grows at a rate proportiona...
 3.1.3.1.5: The radioactive isotope of lead, Pb209, decays at a rate proportio...
 3.1.3.1.6: Initially 100 milligrams of a radioactive substance was present. Af...
 3.1.3.1.7: Determine the halflife of the radioactive substance described in 6.
 3.1.3.1.8: (a) Consider the initialvalue problem dAdt kA, A(0) A0 as the mode...
 3.1.3.1.9: When a vertical beam of light passes through a transparent medium, ...
 3.1.3.1.10: When interest is compounded continuously, the amount of money incre...
 3.1.3.1.11: Archaeologists used pieces of burned wood, or charcoal, found at th...
 3.1.3.1.12: The Shroud of Turin, which shows the negative image of the body of ...
 3.1.3.1.13: A thermometer is removed from a room where the temperature is 70 F ...
 3.1.3.1.14: A thermometer is taken from an inside room to the outside, where th...
 3.1.3.1.15: A small metal bar, whose initial temperature was 20 C, is dropped i...
 3.1.3.1.16: Two large containers A and B of the same size are fille with differ...
 3.1.3.1.17: A thermometer reading 70 F is placed in an oven preheated to a cons...
 3.1.3.1.18: At t 0 a sealed test tube containing a chemical is immersed in a li...
 3.1.3.1.19: A dead body was found within a closed room of a house where the tem...
 3.1.3.1.20: The rate at which a body cools also depends on its exposed surface ...
 3.1.3.1.21: A tank contains 200 liters of fluid in which 30 grams of salt is di...
 3.1.3.1.22: Solve assuming that pure water is pumped into the tank.
 3.1.3.1.23: A large tank is filled to capacity with 500 gallons of pure water. ...
 3.1.3.1.24: In 23, what is the concentration c(t) of the salt in the tank at ti...
 3.1.3.1.25: Solve under the assumption that the solution is pumped out at a fas...
 3.1.3.1.26: Determine the amount of salt in the tank at time t in Example 5 if ...
 3.1.3.1.27: A large tank is partially filled with 100 gallons of flui in which ...
 3.1.3.1.28: In Example 5 the size of the tank containing the salt mixture was n...
 3.1.3.1.29: A 30volt electromotive force is applied to an LRseries circuit in...
 3.1.3.1.30: Solve equation (7) under the assumption that E(t) E0 sin vt and i(0...
 3.1.3.1.31: A 100volt electromotive force is applied to an RCseries circuit in...
 3.1.3.1.32: A 200volt electromotive force is applied to an RCseries circuit i...
 3.1.3.1.33: An electromotive force is applied to an LRseries circuit in which ...
 3.1.3.1.34: Suppose an RCseries circuit has a variable resistor. If the resist...
 3.1.3.1.35: Air Resistance In (14) of Section 1.3 we saw that a differential eq...
 3.1.3.1.36: How High?No Air Resistance Suppose a small cannonball weighing 16 p...
 3.1.3.1.37: How High?Linear Air Resistance Repeat 36, but this time assume that...
 3.1.3.1.38: Skydiving A skydiver weighs 125 pounds, and her parachute and equip...
 3.1.3.1.39: Evaporating Raindrop As a raindrop falls, it evaporates while retai...
 3.1.3.1.40: Fluctuating Population The differential equation dPdt (k cos t)P, w...
 3.1.3.1.41: Population Model In one model of the changing population P(t) of a ...
 3.1.3.1.42: ConstantHarvest Model A model that describes the population of a f...
 3.1.3.1.43: Drug Dissemination A mathematical model for the rate at which a dru...
 3.1.3.1.44: Memorization When forgetfulness is taken into account, the rate of ...
 3.1.3.1.45: Heart Pacemaker A heart pacemaker, shown in Figure 3.1.14, consists...
 3.1.3.1.46: Sliding Box (a) A box of mass m slides down an inclined plane that ...
 3.1.3.1.47: Sliding BoxContinued (a) In let s(t) be the distance measured down ...
 3.1.3.1.48: What Goes Up . . . (a) It is well known that the model in which air...
Solutions for Chapter 3.1: Modeling with FirstOrder Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 3.1: Modeling with FirstOrder Differential Equations
Get Full SolutionsSince 48 problems in chapter 3.1: Modeling with FirstOrder Differential Equations have been answered, more than 20374 students have viewed full stepbystep solutions from this chapter. Chapter 3.1: Modeling with FirstOrder Differential Equations includes 48 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. This expansive textbook survival guide covers the following chapters and their solutions.

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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.

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.

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.