Differential Equations: An Introduction to Modern Methods and Applications
Differential Equations: An Introduction to Modern Methods and Applications
3rd Edition
ISBN: 9781118531778
Author: James R. Brannan, William E. Boyce
Publisher: WILEY
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Chapter 1.1, Problem 11P

Mixing Problems. Many physical systems can be cast in the form of a mixing tank problem. Consider a tank containing a solution—a mixture of solute and solvent–such as salt dissolved in water. Assume that the solution at concentration c i ( t ) flows into a tank at a volume flow rate r i ( t ) and is simultaneously pumped out at a volume flow rate r o ( t ) . If the solution in the tank is well mixed, then the concentration of the outflow is Q ( t ) / V ( t ) , where Q ( t ) is the amount of solute at time t and V ( t ) is the volume of solution in the tank. The differential equation that models the changing amount of solute in the tank is based on the principle of conservation of mass,

d Q d t rate of change of Q ( t ) = c i ( t ) r i ( t ) rate in { Q ( t ) / V ( t ) } r 0 ( t ) rate out , (i)

where V ( t ) also satisfies a mass conservation equation,

d V d t = r i ( t ) r 0 ( t ) . (ii)

If the tank initially contains an amount of solute Q 0 in a volume of solution, V 0 , then initial conditions for Eqs. (i) and (ii) are Q ( 0 ) = Q 0 and V ( 0 ) = V 0 , respectively.

A tank contains 100 gallons (gal) of water and 50 ounces (oz) of salt. Water containing a salt concentration of 1 4 ( 1 + 1 2 sin t ) oz/gal flows into the tank at a rate of 2 gal/min , and the mixture flows out at same rate. Write initial value problem for the amount of salt in the tank at any time t .

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Chapter 1 Solutions

Differential Equations: An Introduction to Modern Methods and Applications

Ch. 1.1 - Mixing Problems. Many physical systems can be cast...Ch. 1.1 - Mixing Problems. Many physical systems can be cast...Ch. 1.1 - Pharmacokinetics. A simple model for the...Ch. 1.1 - A certain drug is being administered intravenously...Ch. 1.1 - Continuously Compounded Interest. The amount of...Ch. 1.1 - Continuously Compounded Interest. The amount of...Ch. 1.1 - Continuously Compounded Interest. The amount of...Ch. 1.1 - Continuously Compounded Interest. The amount of...Ch. 1.1 - A spherical raindrop evaporates at a rate...Ch. 1.1 - Prob. 20PCh. 1.2 - Phase Line Diagrams. Problems through involve...Ch. 1.2 - Phase Line Diagrams. Problems 1 through 7 involve...Ch. 1.2 - Phase Line Diagrams. Problems through involve...Ch. 1.2 - Phase Line Diagrams. Problems 1 through 7 involve...Ch. 1.2 - Phase Line Diagrams. Problems through involve...Ch. 1.2 - Phase Line Diagrams. Problems 1 through 7 involve...Ch. 1.2 - Phase Line Diagrams. Problems through involve...Ch. 1.2 - Problems 8 through 13 involve equations of the...Ch. 1.2 - Problems 8 through 13 involve equations of the...Ch. 1.2 - Problems 8 through 13 involve equations of the...Ch. 1.2 - Problems 8 through 13 involve equations of the...Ch. 1.2 - Problems 8 through 13 involve equations of the...Ch. 1.2 - Problems through involve equations of the form ....Ch. 1.2 - Direction Fields. In each of problems 14 through...Ch. 1.2 - Direction Fields. In each of problems through...Ch. 1.2 - Direction Fields. In each of problems 14 through...Ch. 1.2 - Direction Fields. In each of problems through...Ch. 1.2 - Direction Fields. In each of problems 14 through...Ch. 1.2 - Direction Fields. In each of problems through...Ch. 1.2 - In each of problems through draw a direction...Ch. 1.2 - In each of problems 20 through 23 draw a direction...Ch. 1.2 - In each of problems through draw a direction...Ch. 1.2 - In each of problems through draw a direction...Ch. 1.2 - Consider the following list of differential...Ch. 1.2 - Consider the following list of differential...Ch. 1.2 - Consider the following list of differential...Ch. 1.2 - Consider the following list of differential...Ch. 1.2 - Consider the following list of differential...Ch. 1.2 - Consider the following list of differential...Ch. 1.2 - Verify that the function in Eq.(11) is a solution...Ch. 1.2 - Show that Asint+Bcost=Rsin(t), where R=A2+B2 and ...Ch. 1.2 - If in the exponential model for population growth,...Ch. 1.2 - An equation that is frequently used to model the...Ch. 1.2 - In addition to the Gompertz equation (see Problem...Ch. 1.2 - A chemical of fixed concentration flows into a...Ch. 1.2 - A pond forms as water collects in a conical...Ch. 1.2 - The Solow model of economic growth (ignoring the...Ch. 1.3 - In each of Problems through , determine the order...Ch. 1.3 - In each of Problems 1 through 6, determine the...Ch. 1.3 - In each of Problems 1 through 6, determine the...Ch. 1.3 - In each of Problems through , determine the order...Ch. 1.3 - In each of Problems through , determine the order...Ch. 1.3 - In each of Problems through , determine the order...Ch. 1.3 - Show that Eq. (10) can be matched to each equation...Ch. 1.3 - Show that Eq. (10) can be matched to each equation...Ch. 1.3 - Show that Eq. (10) can be matched to each equation...Ch. 1.3 - Show that Eq. (10) can be matched to each equation...Ch. 1.3 - Show that Eq. (10) can be matched to each equation...Ch. 1.3 - Show that Eq. (10) can be matched to each equation...Ch. 1.3 - In each of Problems 13 through 20, verify that...Ch. 1.3 - In each of Problems through , verify that each...Ch. 1.3 - In each of Problems through , verify that each...Ch. 1.3 - In each of Problems 13 through 20, verify that...Ch. 1.3 - In each of Problems 13 through 20, verify that...Ch. 1.3 - In each of Problems through , verify that each...Ch. 1.3 - In each of Problems 13 through 20, verify that...Ch. 1.3 - In each of Problems through , verify that each...Ch. 1.3 - In each of Problems 21 through 24, determine the...Ch. 1.3 - In each of Problems through , determine the...Ch. 1.3 - In each of Problems through , determine the...Ch. 1.3 - In each of Problems 21 through 24, determine the...Ch. 1.3 - In each of Problems 25 and 26, determine the...Ch. 1.3 - In each of Problems 25 and 26, determine the...Ch. 1.3 - In Problems 27 through 31, verify that y(t)...Ch. 1.3 - In Problems through , verify that satisfies the...Ch. 1.3 - In Problems through , verify that satisfies the...Ch. 1.3 - In Problems 27 through 31, verify that y(t)...Ch. 1.3 - In Problems through , verify that satisfies the...Ch. 1.3 - Verify that the function (t)=c1et+c2e2t is a...Ch. 1.3 - Verify that the function is a solution of the...Ch. 1.3 - Verify that the function (t)=c1etcos2t+c2etsin2t...
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