Modelling with Differential Equations Modelling with Differential - PowerPoint PPT Presentation

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 concentration/mass/volume of a fluid/element/product Modelling with Differential Equations

◮ Problems with inflow/outflow ◮ Equation for concentration/mass/volume of a fluid/element/product rate of change = rate in − rate out ◮ Modelling with Differential Equations

Problem #5 in Section 2.3 Modelling with Differential Equations

Problem #5 in Section 2.3 ◮ A tank contains 100 gal of water and 50 oz of salt. Modelling with Differential Equations

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

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

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

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

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

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

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

Model Modelling with Differential Equations

Model ◮ y ( t ) amount of salt at time t Modelling with Differential Equations

Model ◮ y ( t ) amount of salt at time t ◮ y (0) = 50 Modelling with Differential Equations

Model ◮ y ( t ) amount of salt at time t ◮ y (0) = 50 ◮ Volume = constant = 100. ◮ Equation Modelling with Differential Equations

Model ◮ y ( t ) amount of salt at time t ◮ y (0) = 50 ◮ Volume = constant = 100. ◮ Equation dy ( t ) = 1 � 1 + 1 � − 1 2 sin( t ) 50 y ( t ) 2 dt Modelling with Differential Equations

Model ◮ y ( t ) amount of salt at time t ◮ y (0) = 50 ◮ Volume = constant = 100. ◮ Equation dy ( t ) = 1 � 1 + 1 � − 1 2 sin( t ) 50 y ( t ) 2 dt ◮ Solution Modelling with Differential Equations

Model ◮ y ( t ) amount of salt at time t ◮ y (0) = 50 ◮ Volume = constant = 100. ◮ Equation dy ( t ) = 1 � 1 + 1 � − 1 2 sin( t ) 50 y ( t ) 2 dt ◮ Solution y ( t ) = 63150 5002 sin( t ) − 625 25 2501 e − t / 50 + 25 + 2501 cos( t ) Modelling with Differential Equations

Plot of y ( t ) Modelling with Differential Equations

Plot of y ( t ) 50 45 40 f 35 30 25 0 200 400 600 800 1000 x Modelling with Differential Equations

Plot of y ( t ) 50 45 40 f 35 30 25 0 200 400 600 800 1000 x Modelling with Differential Equations

Problem #5 in Section 2.3 Modelling with Differential Equations

Problem #5 in Section 2.3 ◮ A tank contains 100 gal of water and 50 oz of salt. Modelling with Differential Equations

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

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

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

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

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

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

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