Modelling with Differential Equations

Modelling with Differential Equations

I Problems with inflow/outflow

I Equation for concentration/mass/volume of afluid/element/product

I rate of change = ratein − rateout

Modelling with Differential Equations

I Problems with inflow/outflow

I Equation for concentration/mass/volume of afluid/element/product

I rate of change = ratein − rateout

Modelling with Differential Equations

I Problems with inflow/outflow

I Equation for concentration/mass/volume of afluid/element/product

I rate of change = ratein − rateout

Modelling with Differential Equations

I Problems with inflow/outflow

I Equation for concentration/mass/volume of afluid/element/product

I rate of change = ratein − rateout

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level.

What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level?

What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Model

I y(t) amount of salt at time t

I y(0) = 50

I Volume = constant = 100.

I Equation

dy(t)

dt=

1

2

(1 +

1

2sin(t)

)− 1

50y(t)

I Solution

y(t) =63150

2501e−t/50 + 25 +

25

5002sin(t) − 625

2501cos(t)

Modelling with Differential Equations

Model

I y(t) amount of salt at time t

I y(0) = 50

I Volume = constant = 100.

I Equation

dy(t)

dt=

1

2

(1 +

1

2sin(t)

)− 1

50y(t)

I Solution

y(t) =63150

2501e−t/50 + 25 +

25

5002sin(t) − 625

2501cos(t)

Modelling with Differential Equations

Model

I y(t) amount of salt at time t

I y(0) = 50

I Volume = constant = 100.

I Equation

dy(t)

dt=

1

2

(1 +

1

2sin(t)

)− 1

50y(t)

I Solution

y(t) =63150

2501e−t/50 + 25 +

25

5002sin(t) − 625

2501cos(t)

Modelling with Differential Equations

Model

I y(t) amount of salt at time t

I y(0) = 50

I Volume = constant = 100.

I Equation

dy(t)

dt=

1

2

(1 +

1

2sin(t)

)− 1

50y(t)

I Solution

y(t) =63150

2501e−t/50 + 25 +

25

5002sin(t) − 625

2501cos(t)

Modelling with Differential Equations

Model

I y(t) amount of salt at time t

I y(0) = 50

I Volume = constant = 100.

I Equation

dy(t)

dt=

1

2

(1 +

1

2sin(t)

)− 1

50y(t)

I Solution

y(t) =63150

2501e−t/50 + 25 +

25

5002sin(t) − 625

2501cos(t)

Modelling with Differential Equations

Model

I y(t) amount of salt at time t

I y(0) = 50

I Volume = constant = 100.

I Equation

dy(t)

dt=

1

2

(1 +

1

2sin(t)

)− 1

50y(t)

I Solution

y(t) =63150

2501e−t/50 + 25 +

25

5002sin(t) − 625

2501cos(t)

Modelling with Differential Equations

Model

I y(t) amount of salt at time t

I y(0) = 50

I Volume = constant = 100.

I Equation

dy(t)

dt=

1

2

(1 +

1

2sin(t)

)− 1

50y(t)

I Solution

y(t) =63150

2501e−t/50 + 25 +

25

5002sin(t) − 625

2501cos(t)

Modelling with Differential Equations

Plot of y(t)

0 200 400 600 800 1000

2530

3540

4550

x

f

Modelling with Differential Equations

Plot of y(t)

0 200 400 600 800 1000

2530

3540

4550

x

f

Modelling with Differential Equations

Plot of y(t)

0 200 400 600 800 1000

2530

3540

4550

x

f

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level.

What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level?

What is theamplitude of the oscillation.

Modelling with Differential Equations

Problem #5 in Section 2.3

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

I Water containing a salt concentration of14 (1 + 1

2 sin(t)) oz/gal flows into the tank at a rate of2 gal/min and

I 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 seethe ultimate behavior of the graph.

(c) The long time behavior of the solution is an oscillation about acertain constant level. What is this level? What is theamplitude of the oscillation.

Modelling with Differential Equations