Explanation Given: The linear system of differential equations is, d x d t = κ V A ( y − x ) d y d t = κ V B ( x − y ) x 0 > y 0 > 0 , κ > 0 and x ( t ) > y ( t ) Calculation: The linear system of differential equations is, d x d t = κ V A ( y − x ) d y d t = κ V B ( x − y ) Here, V A , V B are the volumes if compartments A and B and κ is the permeability factor. Substitute k 1 for κ V A and k 2 for κ V B in above equations. d x d t = k 1 ( y − x ) d y d t = k 2 ( x − y ) Assume z ( t ) = x ( t ) − y ( t ) d z d t = d x d t − d y d t = k 1 ( y − x ) − k 2 ( x − y ) = k 1 ( − z ) − k 2 ( z ) d z d t + ( k 1 + k 2 ) z = 0 Separate the variables of the above equation. d z z = − ( k 1 + k 2 ) d t Integrate the above equation. ∫ d z z = ∫ − ( k 1 + k 2 ) d t ln | z | = − ( k 1 + k 2 ) t + c z ( t ) = c 1 e − ( k 1 + k 2 ) t (1) d x d t = − k 1 ( y − x ) = k 1 z = c 1 k 1 e − ( k 1 + k 2 ) t Separate the variables of the above equation. d x = c 1 k 1 e − ( k 1 + k 2 ) t d t Integrate the above equation. ∫ d x = ∫ c 1 k 1 e − ( k 1 + k 2 ) t d t x ( t ) = c 1 k 1 k 1 + k 2 . e − ( k 1 + k 2 ) t + c 2 (2) Since z ( t ) = x ( t ) − y ( t ) , y ( t ) = x ( t ) − z ( t ) y = c 1 k 1 k 1 + k 2 . e − ( k 1 + k 2 ) t − c 1 e − ( k 1 + k 2 ) t = c 1 k 2 k 1 + k 2 . e − ( k 1 + k 2 ) t + c 2 Since, x ( 0 ) = x 0 , y ( 0 ) = y 0 z ( 0 ) = x ( 0 ) − y ( 0 ) = x 0 − y 0 Substitute 0 for t and x 0 − y 0 for z ( 0 ) in equation (1)

Bundle: A First Course in Differential Equations with Modeling Applications, Loose-leaf Version, 11th + WebAssign Printed Access Card for Zill's A ... Applications, 11th Edition, Single-Term

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ISBN: 9781337604994

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 3.3, Problem 20E

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Chapter 3.3, Problem 19E

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Chapter 3 Solutions

Bundle: A First Course in Differential Equations with Modeling Applications, Loose-leaf Version, 11th + WebAssign Printed Access Card for Zill's A ... Applications, 11th Edition, Single-Term

Ch. 3.1 - The population of a community is known to increase...Ch. 3.1 - Suppose it is known that the population of the...Ch. 3.1 - The population of a town grows at a rate...Ch. 3.1 - The population of bacteria in a culture grows at a...Ch. 3.1 - The radioactive isotope of lead, Pb-209, decays at...Ch. 3.1 - Initially 100 milligrams of a radioactive...Ch. 3.1 - Determine the half-life of the radioactive...Ch. 3.1 - Consider the initial-value problem dA/dt = kA,...Ch. 3.1 - When a vertical beam of light passes through a...Ch. 3.1 - When interest is compounded continuously, the...

Ch. 3.1 - Carbon Dating Archaeologists used pieces of burned...Ch. 3.1 - The Shroud of Turin, which shows the negative...Ch. 3.1 - Newtons Law of Cooling/Warming A thermometer is...Ch. 3.1 - A thermometer is taken from an inside room to the...Ch. 3.1 - A small metal bar, whose initial temperature was...Ch. 3.1 - Two large containers A and B of the same size are...Ch. 3.1 - A thermometer reading 70 F is placed in an oven...Ch. 3.1 - At t = 0 a sealed test tube containing a chemical...Ch. 3.1 - A dead body was found within a closed room of a...Ch. 3.1 - The rate at which a body cools also depends on its...Ch. 3.1 - A tank contains 200 liters of fluid in which 30...Ch. 3.1 - Solve Problem 21 assuming that pure water is...Ch. 3.1 - A large tank is filled to capacity with 500...Ch. 3.1 - In Problem 23, what is the concentration c(t) of...Ch. 3.1 - Solve Problem 23 under the assumption that the...Ch. 3.1 - Determine the amount of salt in the tank at time t...Ch. 3.1 - A large tank is partially filled with 100 gallons...Ch. 3.1 - In Example 5 the size of the tank containing the...Ch. 3.1 - A 30-volt electromotive force is applied to an...Ch. 3.1 - Prob. 30E Ch. 3.1 - A 100-volt electromotive force is applied to an...Ch. 3.1 - A 200-volt electromotive force is applied to an...Ch. 3.1 - An electromotive force E(t)={120,0t200,t20 is...Ch. 3.1 - An LR-series circuit has a variable inductor with...Ch. 3.1 - Air Resistance In (14) of Section 1.3 we saw that...Ch. 3.1 - How High?No Air Resistance Suppose a small...Ch. 3.1 - How High?Linear Air Resistance Repeat Problem 36,...Ch. 3.1 - Skydiving A skydiver weighs 125 pounds, and her...Ch. 3.1 - Rocket Motion Suppose a small single-stage rocket...Ch. 3.1 - Rocket MotionContinued In Problem 39 suppose of...Ch. 3.1 - Evaporating Raindrop As a raindrop falls, it...Ch. 3.1 - Prob. 42E Ch. 3.1 - Prob. 43E Ch. 3.1 - Constant-Harvest model A model that describes the...Ch. 3.1 - Drug Dissemination A mathematical model for the...Ch. 3.1 - Memorization When forgetfulness is taken into...Ch. 3.1 - Heart Pacemaker A heart pacemaker, shown in Figure...Ch. 3.1 - Sliding Box (a) A box of mass m slides down an...Ch. 3.1 - Sliding Box—Continued In Problem 48 let s(t) be...Ch. 3.1 - What Goes Up (a) It is well known that the model...Ch. 3.2 - The number N(t) of supermarkets throughout the...Ch. 3.2 - The number N(t) of people in a community who are...Ch. 3.2 - A model for the population P(t) in a suburb of a...Ch. 3.2 - Census data for the United States between 1790 and...Ch. 3.2 - (a) If a constant number h of fish are harvested...Ch. 3.2 - Investigate the harvesting model in Problem 5 both...Ch. 3.2 - Repeat Problem 6 in the case a = 5, b = 1, h = 7.Ch. 3.2 - (a) Suppose a = b = 1 in the Gompertz differential...Ch. 3.2 - Two chemicals A and B are combined to form a...Ch. 3.2 - Solve Problem 9 if 100 grams of chemical A is...Ch. 3.2 - Leaking cylindrical tank A tank in the form of a...Ch. 3.2 - Leaking cylindrical tank—continued When friction...Ch. 3.2 - Leaking Conical Tank A tank in the form of a...Ch. 3.2 - Inverted Conical Tank Suppose that the conical...Ch. 3.2 - Air Resistance A differential equation for the...Ch. 3.2 - How High?Nonlinear Air Resistance Consider the...Ch. 3.2 - That Sinking Feeling (a) Determine a differential...Ch. 3.2 - Solar Collector The differential equation...Ch. 3.2 - Tsunami (a) A simple model for the shape of a...Ch. 3.2 - Evaporation An outdoor decorative pond in the...Ch. 3.2 - Doomsday equation Consider the differential...Ch. 3.2 - Doomsday or extinction Suppose the population...Ch. 3.2 - Skydiving A skydiver is equipped with a stopwatch...Ch. 3.2 - Prob. 27E Ch. 3.2 - Old Man River In Figure 3.2.8(a) suppose that the...Ch. 3.2 - Prob. 29E Ch. 3.2 - Prob. 30E Ch. 3.2 - Prob. 31E Ch. 3.2 - Prob. 32E Ch. 3.2 - Time Drips By The clepsydra, or water clock, was a...Ch. 3.2 - (a) Suppose that a glass tank has the shape of a...Ch. 3.2 - Prob. 35E Ch. 3.3 - We have not discussed methods by which systems of...Ch. 3.3 - In Problem 1 suppose that time is measured in...Ch. 3.3 - Use the graphs in Problem 2 to approximate the...Ch. 3.3 - Construct a mathematical model for a radioactive...Ch. 3.3 - Potassium-40 Decay The chemical element potassium...Ch. 3.3 - Potassium-Argon Dating The knowledge of how K-40...Ch. 3.3 - Consider two tanks A and B, with liquid being...Ch. 3.3 - Use the information given in Figure 3.3.6 to...Ch. 3.3 - Two very large tanks A and B are each partially...Ch. 3.3 - Three large tanks contain brine, as shown in...Ch. 3.3 - Consider the Lotka-Volterra predator-prey model...Ch. 3.3 - Show that a system of differential equations that...Ch. 3.3 - Determine a system of first-order differential...Ch. 3.3 - Prob. 16E Ch. 3.3 - SIR Model A communicable disease is spread...Ch. 3.3 - Prob. 18E Ch. 3.3 - Prob. 19E Ch. 3.3 - Prob. 20E Ch. 3.3 - Mixtures Solely on the basis of the physical...Ch. 3.3 - Newtons Law of Cooling/Warming As shown in Figure...Ch. 3 - Answer Problems 1 and 2 without referring back to...Ch. 3 - Answer Problems 1 and 2 without referring back to...Ch. 3 - Prob. 3RE Ch. 3 - Air containing 0.06% carbon dioxide is pumped into...Ch. 3 - tzi the Iceman In September of 1991 two German...Ch. 3 - Prob. 6RE Ch. 3 - Prob. 7RE Ch. 3 - Suppose a cell is suspended in a solution...Ch. 3 - Suppose that as a body cools, the temperature of...Ch. 3 - According to Stefans law of radiation the absolute...Ch. 3 - Suppose an RC-series circuit has a variable...Ch. 3 - A classical problem in the calculus of variations...Ch. 3 - A model for the populations of two interacting...Ch. 3 - Initially, two large tanks A and B each hold 100...Ch. 3 - Prob. 15RE Ch. 3 - When all the curves in a family G(x, y, c1) = 0...Ch. 3 - Prob. 17RE Ch. 3 - Prob. 18RE Ch. 3 - Prob. 19RE Ch. 3 - Sawing Wood A long uniform piece of wood (cross...Ch. 3 - Solve the initial-value problem in Problem 20 when...Ch. 3 - Prob. 22RE

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