modelling with di erential equations - university of notre...
TRANSCRIPT
![Page 1: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/1.jpg)
Modelling with Differential Equations
Modelling with Differential Equations
![Page 2: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/2.jpg)
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
![Page 3: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/3.jpg)
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
![Page 4: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/4.jpg)
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
![Page 5: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/5.jpg)
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
![Page 6: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/6.jpg)
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
![Page 7: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/7.jpg)
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
![Page 8: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/8.jpg)
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
![Page 9: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/9.jpg)
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
![Page 10: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/10.jpg)
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
![Page 11: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/11.jpg)
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
![Page 12: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/12.jpg)
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
![Page 13: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/13.jpg)
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
![Page 14: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/14.jpg)
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
![Page 15: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/15.jpg)
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
![Page 16: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/16.jpg)
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
![Page 17: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/17.jpg)
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
![Page 18: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/18.jpg)
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
![Page 19: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/19.jpg)
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
![Page 20: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/20.jpg)
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
![Page 21: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/21.jpg)
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
![Page 22: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/22.jpg)
Plot of y(t)
0 200 400 600 800 1000
2530
3540
4550
x
f
Modelling with Differential Equations
![Page 23: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/23.jpg)
Plot of y(t)
0 200 400 600 800 1000
2530
3540
4550
x
f
Modelling with Differential Equations
![Page 24: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/24.jpg)
Plot of y(t)
0 200 400 600 800 1000
2530
3540
4550
x
f
Modelling with Differential Equations
![Page 25: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/25.jpg)
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
![Page 26: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/26.jpg)
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
![Page 27: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/27.jpg)
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
![Page 28: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/28.jpg)
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
![Page 29: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/29.jpg)
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
![Page 30: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/30.jpg)
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
![Page 31: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/31.jpg)
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
![Page 32: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/32.jpg)
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
![Page 33: Modelling with Di erential Equations - University of Notre ...aache/slides-20580-sp2017/april05.pdfProblem #5 in Section 2.3 I A tank contains 100 gal of water and 50 oz of salt. I](https://reader033.vdocuments.net/reader033/viewer/2022042301/5ecc0b7626fdee54ff16452f/html5/thumbnails/33.jpg)
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