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Differential Equations and Linear Algebra 2nd edition Jerry Farlow, James E. Hall, Jean Marie McDill, Beverly H. West - Solutions

A 1960 New York Times article announced: Archaeologists Claim Sumerian Civilization Occupied the Tigris Valley 5.000 Years Ago. Assuming the archaeologists used Carbon- 14 to date the site, determine the percentage of Carbon-14 found in the relevant samples.
Solve the homogeneous equation dx/dt + 2x/(100-t) = 0 Corresponding to the DE in Example 2 using separation of variables
Fresh water is poured at a rate of 2 gal / min into a tank A, which initially contains 100 gal of a salt solution with concentration 0.5 lb / gal. The stirred mixture flows out of tank A at the same rate and into a second tank B that initially contained 100 gal of fresh water. The mixture in tank B
A cascade of several tanks is shown in Fig. Initially, tank 0 contains 1 gal of alcohol and 1 gal of water, while the other tanks contain 2 gal of pure water. Fresh water is pumped into tank O al the rate of I gal/min, and the varying mixture in each tank is pumped into the next tank at the same
Consider the cascading arrangement of tanks shown in Fig. 2.4.10, with Vi = 200 gal.V2 = 200 gal, and V3 = 500 gal as the volumes of brine in the three tanks. Each tank initially contains 20 lb of salt. The inflow rates and outflow rates for tanks 1 and 2 are all 5 gal / sec, but the outflow rate
Instead of using the theory of Sec. 2.1, solve the cooling/heating problemBy separation of variables does this seem easier than the Euler-Lagrange approach?
Solve Newton's equation dT/dt = k(M-T) by making the change of variable y = T - M to transform it into the decay equation.
At noon, with the temperature in your house at 75°F and the outside temperature at 95°F, your air conditioner breaks down. Suppose that the time constant 1 / k for your house is 4 hours. (a) What will the temperature in your house be at 2:00 PM? (b) When will the temperature in your house reach
Suppose that it is 70° F in your house when the furnace breaks down at midnight. The outside temperature is 10" F. You notice that after 30 minutes the inside temperature has dropped 10 50° F. (a) What will the temperature be after one hour (that is. at 1:00 AM)? (b) How long will it take for the
The rate at which a drug is absorbed into the bloodstream is modeled by the first-order differential equation dC/dt = a - bC(t) Where a and b are positive constants and C(t) denotes the concentration of drug in the bloodstream at time 1. Assuming that no drug is initially present in the
A cold beer with an initial temperature of 35°F warms up to 40°F in 10 minutes while sitting in a room with temperature 70°F. What will the temperature of the beer be after t minutes? After 20 minutes?
John and Maria are having dinner, and each orders a cup of coffee. John cools his coffee with some cream. They wait I 0 minutes and then Maria cools her coffee with the same amount of cream. The two then begin to drink. Who drinks the hotter coffee?
Initia11y. 50 lb of salt is dissolved in a tank containing 300 gal of water. A sa1t solution with 2 lb/gal concentration is poured into the tank at 3 gal / min. The mixture, after stirring, flows from the tank at the same rate the brine is entering the tank. (a) Find the amount of salt in the tank
Professor Farlow always has a cup of coffee before his 8:00 AM differential equations class. Suppose the coffee is 200° F when poured from the coffee pot at 7:30 AM, and 15 minutes later it cools to 1 20°F in a room whose temperature is 70°F. However, Professor Farlow never drinks his coffee
At 1:00 PM, Carlos puts into the refrigerator a can of soda, which has been standing out in a room with temperature 70°F. The temperature inside the refrigerator is 40°F. Fifteen minutes later, at 1:15 PM, the temperature of the soda has fallen to 60°F. At some later time Carlos removes the soda
Initially a 100-liter tank contains a salt solution with concentration 0.5 kg/liter. A fresher solution with concentration 0.1 kg / liter flows into the tank at the rate of 4 liter / min. The contents of the tank are kept well stirred and the mixture flows out at the same rate it flows in. (a) Find
At the start 5 lb of salt are dissolved in 20 gal of water. Salt solution with concentration 2 lb / gal is added at a rate of 3 gal / min. and the well-stirred mixture is drained out at the same rate of flow. How long should this process continue to raise the amount of salt in the tank to 25 lb?
A tank initially contains 200 gallons of fresh water but then a salt solution of unknown concentration is poured into the tank at 2 gal / min. The well-stirred mixture flows out of the tank at the same rate. After 1 20 min, the concentration of salt in the tank is 1.4 lb/gal. What is the
A 600-gaJlon tank is filled with 300 gal of pure water. A spigot is opened and a sa1t solution containing 1 lb of sa1t per gallon of solution begins flowing into the tank at a rate of 3 gal/min. Simultaneously a drain is opened at the bottom of the tank a1lowing the solution to leave the tank at a
Lake Erie has a volume of roughly 100 cubic miles and its equa1 inflow and outflow rates are 40 cubic miles per year. At year t = 0 a certain pollutant has a volume concentration of 0.05% but after that t h e concentration of pollutant flowing into the lake drops to 0.01 %. Answer the following
Into a tank containing 100 gal of fresh water Wei Chen was to have added 10 lb of salt but accidentally added 20 lb instead. To correct her mistake she started adding fresh water at a rate of 3 gal / min. while drawing off well-mixed solution at the same rate. How long will it take until the tank
A 1,000-gallon tank contains 200 gal of pure water. A brine solution containing .I lb of salt per gal is flowing into the tank at a rate of 4 gal / sec, and the well-stirred mixture is leaving the tank at the same rate. Let x denote the amount of salt in the tank at time t.(a) Set u p (but d o not
In Problems, use direction fields and isoclines to make a qualitative sketch of the solutions, determine the equilibrium values (when they exist), and classify them as stable or unstable. Consider the parameters a and b to be positive in each case a. y' = ay + by2 b. y' = ay - by2
Inflection Points For many DEs, the easiest way to pinpoint inflection points is not from the solution but from the DE itself. Find y" by differentiating y', remembering to use the chain rule wherever y occurs then substitute for y' from the DE and set y" = 0. Solve for y to find the inflection
A population grows according to the logistic law with a limiting population of 5 x 109 individuals. The initial population of 0.2 x 109 begins growing by doubling every hour. What will the population be after 4 hours?
Suppose that we start at time to = 0 with a sample of 1000 cells. One day later we see that the population has doubled, and some time later we notice that the population has stabilized at 100000. Assume a logistic growth model. (a) What is the population after 5 days? (b) How long does it take the
If the growth of a population follows the logistic model but is subject to "harvesting" (such as hunting or fishing), the model Becomes(a) Show that when the harvest rate h is constant, a maximum sustainable harvest hmax is rL/4, which occurs when the population is half the carrying capacity. (b)
A certain piece of dubious information about the cancellation of final exams began to spread one day on a college campus with a population of 80,000 students. Assume that initially one thousand students heard the rumor on the radio. Within a day l0.000 students had heard the rumor. Assume that the
A rumor about dihydrogen monoxide in the drinking water began to spread one day i n a city o f population 200,000. After one week, 1,000 people had been alarmed by the news. Assume that the rate of increase of the number N of people who have heard the rumor is proportional to the product of those
Illustrate the semi-stability of the equilibrium point of dy/dt = (1-y)2. y ≥ 0, by making a phase line and sketching typical solution curves.
The Gompertz equation dy/dt = y(a-b ln y) where a and b are parameters, is used in actuarial studies, and to model growth of objects as diverse as tumors and organizations. (a) Show that the solution to the Gompertz equation is y(t) = ea/bece-bt (b) Solve the IVP for this equation with y(0) =
Autonomous Analysis For the first-order autonomous equations in Problems, complete the following. (a) Sketch qualitative solution graphs. (b) Highlight the equilibrium points of the equation and draw phase-line arrows along the y-axis indicating the increasing or decreasing behavior of the solution
According to Stefan's Law of Radiation (previously examined in Sec. 1.3, Problem 55 and Sec. 1.4, Problem 11) the rate of change of the radiation energy of a body at absolute temperature T is given by dT /dt = k(M4 - T4), where k > 0 and M is the ambient or surrounding absolute temperature. Sketch
Refer to the final subsection "Logistic Model in Another Context" and Fig. 2.5.9 to explore the application of the logistic equation to the oil industry, as follows.(a) The phrase "Hubbert peak" refers to a graph not shown, the graph of y' versus t. Sketch this missing graph from the information in
Solve the logistic IVP y' = ky (l - y), y(0) = y0. By means of the change of variable z = y/( l - y). Solve the resulting DE in z = z(t) and then re-substitute to obtain y(t). (Your result should be equation (10) with L = 1)
Two chemicals A and B react to form the substance X according to the law dx /dt = k(100 - x) (50- x), where k is a positive constant and x is the amount of substance X. Describe the amount of substance X, given the initial conditions in (a), (b), and (c). Sketch the direction field, equilibrium
Make the change of variable t = -r in the threshold IVP (14) and verify that this results in the IVP for the logistic equation. Use the solution of the logistic problem to verify the solution (15).
(a) Show that if y0 (b) Show that if y0 > T the solution y(r) of (l4) "blows up" at time
For the differential equation dy/dt = ay - y3,Show that 0 is a bifurcation point of the parameter a as follows(a) Show that if a ‰¤ 0 there is only one equilibrium point at 0 and it is stable.(b) Show that if a > 0 there are three equilibrium points: 0, which is unstable, and ±
Study the relationship between the values of the parameter b in the differential equation dy /dt = y2 + by + I and the equilibrium points of the equation and their stability. (a) Show that for lbl > 2 there are two equilibrium points; for |b| = 2, one; and for |b| < 2, none. (b) Determine the
In Problem we study the effect of parameters on the solutions of differential equations. For each equation, do the following. (a) Determine values of k where the number and / or nature of equilibrium points changes. (b) Draw direction fields and sample solutions to the DE for different values of k.
Four growth equations used by population theorists are given in Problem. Plot solutions for different values of their parameters and try to determine their significance. y' = ky2 + y + 1
Use the same directions for Problems as for Problems 1. Describe in each case what differences are caused by the equations being non autonomous. a. y' = y(y -t) b. y' = (y - 1)2
Consider the systems in Problem(a) Determine and plot the equilibrium points and null-clines for the systems.(b) Show the direction of the vector field between the null-clines as illustrated in Example 2 and Fig. 2.6.4.(c) Sketch some solution curves starring near; but not on, the equilibrium point
We apply the classical predator prey model of Volterrs dx/dt = ax - bxy, dy/dt = -cx + dxy, where x denotes the population of sardines (prey) and y the population of sharks (predators). We now subtract a term from each equation that accounts for the depletion of both species due to external
Consider the competition models for rabbits R and sheep S described in Problem. What are the equailibria what do each signify, and which are stable?dR/dt = R(1200 - 2R - 3S)dS/dt = S(500 - R - S)
The scenarios in Problems de-scribe the interaction of two and three different species of plants or animals. In each case set up a system of differential equations that might be used to model the situation. It is not necessary at this stage to solve the systems. a. We have a population of rabbits
Analyze the models for competition between two species given in Problems, using the following outline. (a) Find and plot the equilibrium points and nullclines. Determine the directions of the vector fields between the nullclines. (b) Decide whether the equilibrium points are unstable (repelling
Consider a situation of two populations competing according to the simpler model dx/dt = x(a - by), dy/dt = y(c -dx). Where a, b, c. and d are positive constants. Find and sketch the equilibrium points and the nullclines, and determine directions of the vector field between the nullclines. Then
For the competition model dx/dt = x(a - bx - cy). dy/dt = y(d - ex - fy), Where the parameters a, b, c. d, e, and f are all positive, the diagrams in Problems show four possible positions of the nullclines and equilibrium points. In each case: (a) Draw arrows in each region between the nullclines
Another model for competition arises when, in the absence of a second species, the first grows logistically and the second exponentially: dx//dt = ax (l -bx.) - cxy, dy/dt = dy - exy. Show that for this model coexistence is impossible.
Basins of Attraction Problems each specify one of the competition scenarios in the previous set of problems. For each stable equilibrium point in the given model, find and color the set of points in the plane whose associated solution curves eventually approach that equilibrium point. An example is
Use these IDE tools to do the following: Analyze the effect of each of the parameters aR. aF. cR, and cF on the system (6). Analyze the effect of harvesting either species at a constant rate. Explain the different outcomes for harvesting predators versus harvesting prey.
Suppose that in the absence of foxes, a rabbit population increases by 15% per year; in the absence of rabbits, a fox population decreases by 25% per year; and in equilibrium, there are 1,000 foxes and 8,000 rabbits. (a) Explain why the Volterra-Lotka equations are dR/dt = 0.15R - 0.00015RF dF/dt =
Do They Compute? Calculate the quantities required in Problems 1-3 whereAnd Or explain why they are undefined. 1. 2A 2. A + 2B 3. 2C - D?
More Multiplication Practice In Problem 1-3, compute the indicated products or explain why it is not possible?1.2. 3.
Which Rules Work for Matrix Multiplication? For each of Problems 1-3, prove the statement in general or give a counterexample. In each problem, A, B, and I denote n x n matrices, and x̅, Y̅. and z̅ denote column vectors in Rn . For the proofs, use the properties of matrix multiplication and do
Find the matrix. Find the nonzero matrices A, B, and C in Problem 1-3. If no such matrix exists, show why?1.2. 3.
Commuters In Problems 1-2, find all the 2 x that commute with the given matrix. 2 matrices.1.Where a ( R 2. Where k ( R, k ( 0 3
Products with Transposes Use matricesAnd To find the indicated products for parts (a) - (d). (a) ATB (b) ABT (c) BTA (d) BAT
Reckoning In Problems 1-3, prove the statements form x matrices A and Band scalars c and d? 1. A - B = A +(-l)B 2. A + B = B + A 3. (c + d)A = cA + dA?
Properties of the Transpose In Problems 1-3, the properties in general using the fact that [a, j]T = [aji], or demonstrate the properties for general 3 ( 3 matrices? 1. (AT)T = A 2. (A + B)T = AT + BT 3. (ka)T = kAT, for any scalar k?
1. Transposes and Symmetry prove that if A is symmetric then so is AT? 2. Symmetry and Products give an example to show that the product of symmetric matrices is not necessarily symmetric?
Constructing symmetry show that for any n x n matrix A, the matrix A + AT is always symmetric?
More Symmetry Demonstrate with an arbitrary 3 ( 2 matrix A that AT A and AAT are always symmetric. (In this case, they are not of the same order.)
Trace of a Matrix Using the following definition, prove the properties of the trace in Problems 1-3.The trace of an n x n matrix A = [aij], denoted Tr A, is the sum of the diagonal elements:1. Tr(A+B) = TrA + TrB 2. Tr(cA) = cTrA 3. Tr (AT) =Tr A?
Complex numbers can serve as entries a matrix just as well real numbers. Compute the expressions in Problem 1-3, whereAnd 1. A + 2B 2. AB 3. BA
Real and Imaginary Components
Square Roots or Zero Are there any 2 ( 2 matrices A, with elements not all zero. satisfying A2 = 0? If so. Give an example. If not, explain?
Zero Divisors If a and b are real or complex numbers such that ab = 0. then either a = 0 or b = 0. Does this property hold for matrices? That is, if A and Bare n ( n matrices such that AB = 0. is if true that we must have A = 0 or B = 0? Prove the result or find a counter example. (Please do not do
Does Cancellation Work? Suppose that AB = A C for matrices A, B, and C. Is it true that B must equal C? Prove the result or find a counterexample?
Tanking Matrices Apart Let A be an n ( n matrix whose jth column is the column vector (n ( 1 matrix) Ai: we can write this as [A1 | A2 | · · · | An) (an example of a partitioned matrix). Let xÌ… be the column vector [x1 x2 ....... xn]T.(a) For(b) Show that in
Diagonal Matrices For Problems 1 and 2. suppose that A and B are n x n diagonal matrices. 1. Show that AB is diagonal? 2. Show that AB = BA?
Upper Triangular Matrices A square n ( n matrix A = [aij] is called upper triangular if aij = 0 for i > j. (All entries below the main diagonal are zero.) (a) Give three examples of upper triangular matrices of different orders. (b) For the 3 ( 3 case, prove that if A and B are both upper
Hard Puzzle the square root of a matrix A is a matrix R such that RR = A. Show that the matrixHas no square root, while the matrix Has an infinite number of square roots.
Orthogonality: In Problems 1-3, find the real values of k for which the given vectors are orthogonal. If there are no such values show why?1.2. 3.
Orthogonality and Subsets In Problems 1-3, find the subset of R3 that is orthogonal to the given vectors. Sketch the subsets in R3?1.2. 3.
Dot Products calculate the dot products of the vectors in Problems 1 - 3. Tell which pairs are orthogonal? 1. [2, 1] ( [- 1, 2] 2. [- 3, 0] ( [2, 1] 3. [2, l, 2] · [3, - 1, 0]
Lengths Show that the distance between the heads of any two vectors uÌ… and vÌ…, as shown in Fig. 3 . 1 .5, has length ||uÌ… - vÌ… ||.
Geometric Vector Operations For1. A + C 2. ½ A + B 3. A - 2B
Triangles for the following pairs of vectors u̅ and v̅ given in Problems 1 and 2, show by can appropriate scalar product whether the triangle formed as in Fig. 3.1.5 is a right triangle sketch the triangle and identify the right angle. Confirm the Pythagorean Theorem. 1. [3, 2], [2, 3] 2. [2, -
Properties of Scalar Products for Problems 1-3, consider the general scalar product on vectors aÌ…, bÌ…. and cÌ… of the same dimension. Either prove the statement to be true or explain why it cannot be true. Take k to be a nonzero scalar.1.2. 3.
Directed Graphs a directed graph is a finite set of points, called nodes, and an associated set of paths or arcs, each connecting two nodes in a given direction. (See Fig. 3. 1 .6.) Think of the arcs as strings or wires that can pass over or under each other; n o actual "contact" takes place except
Tournament Play The directed graph in Fig. 3.1.7 is called a tournament graph because every node is connected to every other node exactly once. The nodes represent players, and an arc from node i to node j stands for the fact that player i has beaten player j. Compute the adjacency matrix T of this
Matrix Vector Form write each system in Problem 1-3 in matrix vector form. Then write the augmented matrix of the system? 1. x + 2y = 1 2x - y = 0 3x + 2y = 1 2. i1 + 2i2 + i3 + 3i4 = 2 I1 - 3i2 + 3i3 = 1 3. r + 2s + r = 1 r - 3s + 3t = 1 4s - 5t = 3
A Special Solution Set in IR3. Write a system of three equations in three unknowns for which the solutions form a single plane in IR3. Determine the parametric equation of the plane?
Reduced Row Echelon Form Determine whether each of the matrices given in Problems 1-3 is in RREF or not. If not explain which condition or conditions fail. Then use elementary row operations to obtain in the RREF.1.2. 3.
Gauss Jordan Reduction use elementary row of operations to reduce each matrix in Problem 1-3 to row echelon form and then to RREF. Then circle the pivot columns in the original matrix?1.2. 3.
Solving System use gauss Jordan reduction to transform the augmented matrix of each system in Problems 1-3 to RREF. Use it to discuss the solutions of the system (i.e., no solutions, a unique solutions, or infinitely many solutions). 1. x + y = 4 x - y = 0 2. y = 2x y = x + 3 3. x + y + z = 0 y + z
Using the Nonhomogeneous Principle determine the solution set W for the associated homogeneous systems in Problems 1-3. Then write the solutions to the systems in the original problems is the form x̅ = x̅h + x̅h, where x̅h ( W? 1. x + y = 4 x - y = 0 2. y = 2x y = x + 3 3. x + y + z = 0 y + z =
The RREF Example consider the systemShow that the RREF of its augmented matrix [A | bÌ…] is given in Example 6?
More Equations Than Variables Consider the systemFind the RREF of the augmented matrix. Determine whether or not the system is consistent. If it is, find the solution or solutions, and give the subset W for the associated homogeneous system?
Homogeneous Systems In Problems 1-3 determine all the solutions of AxÌ… = 0Ì…, where is the matrix shown is the RREF of the augmented matrix [A | bÌ…].a.b. c. [1 - 4 3 0 | 0]
Making Systems Inconsistent: For each of the m ( in Problems n matrices 1-3, determine the rank r of the given matrix A. If r 1.2. 3.
Seeking Consistency: In Problem 1-5, determine the values of k, if any, that would make the augmented matrices shown those of consistent systems. If there is no such k. explain.1.2. 3. 4. 5.
Not Enough Equations: A linear system AxÌ… = bÌ… with fewer equations than unknowns has either infinitely many solutions or no solutions. Examine the augmented matrices [A | bÌ…] and decide which case applies to each matrix?(a)(b)
Not Enough Variables: A linear system Ax̅ = b̅ with fewer unknowns than equations can have infinitely many solutions, no solutions, or a unique solution. Construct the RREFs for the augmented matrices [A I b̅] that illustrate the three possible cases for a system of four equations in two
True/False Questions If true give an explanation. If false, give a counterexample. (a) Different matrices cannot have the same RREF. True or fa1se? (b) If the rank of an m ( n matrix is n. Then the system Ax̅ = b̅ has exactly one solution. True or fa1se? (c) If A is an n ( n matrix and b̅ is a
Equivalence of Systems When one system of equations is obtained from another by a sequence of elementary row operations on their augmented matrices, the systems are equiva1ent (have the same set of solutions) because the transformation can be reversed (hence a solution of the first satisfies the
Homogeneous versus Non-homogeneous: The systems of Problems 32 and 33 differ only in their right-hand sides. Compare their solutions. Explain how this parallels the solution of homogeneous and non-homogeneous linear first order differential equations?

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