Modelling with Differential Equations Modelling with Differential - - PowerPoint PPT Presentation

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Modelling with Differential Equations Modelling with Differential - - PowerPoint PPT Presentation

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Modelling with Differential Equations Modelling with Differential Equations Modelling with Differential Equations Problems with inflow/outflow Modelling with Differential Equations Problems with inflow/outflow Equation for

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Modelling with Differential Equations

Modelling with Differential Equations

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Modelling with Differential Equations

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◮ Problems with inflow/outflow

Modelling with Differential Equations

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◮ Problems with inflow/outflow ◮ Equation for concentration/mass/volume of a

fluid/element/product

Modelling with Differential Equations

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◮ Problems with inflow/outflow ◮ Equation for concentration/mass/volume of a

fluid/element/product

◮

rate of change = ratein − rateout

Modelling with Differential Equations

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Problem #5 in Section 2.3

Modelling with Differential Equations

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Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 oz of salt.

Modelling with Differential Equations

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Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 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

Modelling with Differential Equations

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SLIDE 9

Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 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 of the tank flows out at the same rate.

Modelling with Differential Equations

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SLIDE 10

Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 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 of the tank flows out at the same rate.

(a) Find the amount of salt in the tank at any time

Modelling with Differential Equations

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SLIDE 11

Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 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 of the tank flows out at the same rate.

(a) Find the amount of salt in the tank at any time (b) Plot the solution for a time period long enough so that you see the ultimate behavior of the graph.

Modelling with Differential Equations

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SLIDE 12

Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 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 of the tank flows out at the same rate.

(a) Find the amount of salt in the tank at any time (b) Plot the solution for a time period long enough so that you see the ultimate behavior of the graph. (c) The long time behavior of the solution is an oscillation about a certain constant level.

Modelling with Differential Equations

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Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 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 of the tank flows out at the same rate.

(a) Find the amount of salt in the tank at any time (b) Plot the solution for a time period long enough so that you see the ultimate behavior of the graph. (c) The long time behavior of the solution is an oscillation about a certain constant level. What is this level?

Modelling with Differential Equations

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Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 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 of the tank flows out at the same rate.

(a) Find the amount of salt in the tank at any time (b) Plot the solution for a time period long enough so that you see the ultimate behavior of the graph. (c) The long time behavior of the solution is an oscillation about a certain constant level. What is this level? What is the amplitude of the oscillation.

Modelling with Differential Equations

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Model

Modelling with Differential Equations

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Model

◮ y(t) amount of salt at time t

Modelling with Differential Equations

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Model

◮ y(t) amount of salt at time t ◮ y(0) = 50

Modelling with Differential Equations

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Model

◮ y(t) amount of salt at time t ◮ y(0) = 50 ◮ Volume = constant = 100. ◮ Equation

Modelling with Differential Equations

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Model

◮ y(t) amount of salt at time t ◮ y(0) = 50 ◮ Volume = constant = 100. ◮ Equation

dy(t) dt = 1 2

  • 1 + 1

2 sin(t)

  • − 1

50y(t)

Modelling with Differential Equations

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Model

◮ y(t) amount of salt at time t ◮ y(0) = 50 ◮ Volume = constant = 100. ◮ Equation

dy(t) dt = 1 2

  • 1 + 1

2 sin(t)

  • − 1

50y(t)

◮ Solution

Modelling with Differential Equations

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Model

◮ y(t) amount of salt at time t ◮ y(0) = 50 ◮ Volume = constant = 100. ◮ Equation

dy(t) dt = 1 2

  • 1 + 1

2 sin(t)

  • − 1

50y(t)

◮ Solution

y(t) = 63150 2501 e−t/50 + 25 + 25 5002 sin(t) − 625 2501 cos(t)

Modelling with Differential Equations

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Plot of y(t)

Modelling with Differential Equations

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Plot of y(t)

200 400 600 800 1000 25 30 35 40 45 50 x f

Modelling with Differential Equations

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Plot of y(t)

200 400 600 800 1000 25 30 35 40 45 50 x f

Modelling with Differential Equations

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Problem #5 in Section 2.3

Modelling with Differential Equations

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Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 oz of salt.

Modelling with Differential Equations

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SLIDE 27

Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 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

Modelling with Differential Equations

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Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 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 of the tank flows out at the same rate.

Modelling with Differential Equations

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SLIDE 29

Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 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 of the tank flows out at the same rate.

(a) Find the amount of salt in the tank at any time

Modelling with Differential Equations

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Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 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 of the tank flows out at the same rate.

(a) Find the amount of salt in the tank at any time (b) Plot the solution for a time period long enough so that you see the ultimate behavior of the graph.

Modelling with Differential Equations

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SLIDE 31

Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 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 of the tank flows out at the same rate.

(a) Find the amount of salt in the tank at any time (b) Plot the solution for a time period long enough so that you see the ultimate behavior of the graph. (c) The long time behavior of the solution is an oscillation about a certain constant level.

Modelling with Differential Equations

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Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 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 of the tank flows out at the same rate.

(a) Find the amount of salt in the tank at any time (b) Plot the solution for a time period long enough so that you see the ultimate behavior of the graph. (c) The long time behavior of the solution is an oscillation about a certain constant level. What is this level?

Modelling with Differential Equations

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Problem #5 in Section 2.3

◮ A tank contains 100 gal of water and 50 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 of the tank flows out at the same rate.

(a) Find the amount of salt in the tank at any time (b) Plot the solution for a time period long enough so that you see the ultimate behavior of the graph. (c) The long time behavior of the solution is an oscillation about a certain constant level. What is this level? What is the amplitude of the oscillation.

Modelling with Differential Equations