Math 2250-1Fri Nov 167.1 Systems of differential equations - to model multi-component systems.Examples: 1) input-output models...related to "multi-component modeling", "compartmental analysis"http://en.wikipedia.org/wiki/Multi-compartment_modelHere's a simple 2-tank problem to illustrate the 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1) Find differential equations for solute amounts 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, using input-output modeling. Assume solute concentration is uniform in each tank. If 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 , write down the initial value problem that you expect would have a unique solution.answer (in matrix-vector form):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) Coupled masses and springs (or RLC circuit loops) lead to systems of second order differential equations. For 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2) Use Newton's second law to derive a system of two second order differential equations for 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 the displacements of the respective masses from the equilibrium configuration. What initial value problem do you expect yields unique solutions in this case?Exercise 3) Let 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. Show that in matrix-vector form the system above is equivalent 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 .3) It is always possible to convert a system of higher order differential equations into an equivalent (but larger) system of first order differential equations. This turns out to be important conceptually but also for numerical (computational) methods for solving systems of differential equations. The conversions relies on introducing auxillary functions for intermediate derivatives. For example,Exercise 4) Consider the IVP from Exercise 3:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RKCZwcmltZTtGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHUSYwLjBlbUYnRkMtRkQ2LVEiPUYnRj5GR0ZKRkxGTkZQRlJGVC9GV1EsMC4yNzc3Nzc4ZW1GJy9GWkZqbi1GRDYtUSomdW1pbnVzMDtGJ0Y+RkdGSkZMRk5GUEZSRlQvRldRLDAuMjIyMjIyMmVtRicvRlpGYG8tRjs2JFEiM0YnRj4tRkQ2LVEifkYnRj5GR0ZKRkxGTkZQRlJGVC9GV0ZlbkZZRistRkQ2LVEiK0YnRj5GR0ZKRkxGTkZQRlJGVEZfb0Zhby1GLDYlRi4tRiM2JS1GOzYkUSIyRidGPkYyRjVGQEY+LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnRi9GTUZQLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRiw2LVEoJnByaW1lO0YnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4xMTExMTExZW1GJ0ZERmVuLUYsNi1RIj1GJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGXm9GVEYrLUZHNiVGSS1GIzYlLUZVNiRRIjFGJ0YvRk1GUEZYLUYsNi1RKCZtaW51cztGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjIyMjIyMmVtRicvRkVGW3BGVEYrRkYtRiw2LVEiK0YnRi9GMkY1RjdGOUY7Rj1GP0Zqb0ZccC1GVTYkUSM0MEYnRi9GKy1GSjYlUSRzaW5GJy9GTkY0Ri8tSShtZmVuY2VkR0YkNiQtRiM2Ji1GVTYkUSIzRidGL0YrLUZKNiVRInRGJ0ZNRlBGL0YvRi8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUY7NiRGQkY+Rj5GPi1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZQLyUpc3RyZXRjaHlHRlAvJSpzeW1tZXRyaWNHRlAvJShsYXJnZW9wR0ZQLyUubW92YWJsZWxpbWl0c0dGUC8lJ2FjY2VudEdGUC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmluLUYsNiUtRi82JVEiYkYnRjJGNUY4RkAtRks2LVEiLEYnRj5GTi9GUkY0RlNGVUZXRllGZW4vRmhuUSYwLjBlbUYnL0Zbb1EsMC4zMzMzMzMzZW1GJy1GSzYtUSJ+RidGPkZORlFGU0ZVRldGWUZlbkZlby9GW29GZm9GKy1GSzYtUSgmcHJpbWU7RidGPkZORlFGU0ZVRldGWUZlbi9GaG5RLDAuMTExMTExMWVtRidGXHBGQ0ZKLUYsNiVGXm8tRiM2JS1GOzYkUSIyRidGPkYyRjVGQEY+LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnRi9GTUZQLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GVTYkRlpGL0YvRi8tRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZgby1GRzYlLUZKNiVRImNGJ0ZNRlAtRiM2JS1GVTYkUSIxRidGL0ZNRlBGWC1GLDYtUSIsRidGL0YyL0Y2Rk9GN0Y5RjtGPUY/RkEvRkVRLDAuMzMzMzMzM2VtRidGK0ZGLUYsNi1RKCZwcmltZTtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMTExMTExMWVtRidGREZlbkZcby1GRzYlRmRvRlJGWEYv . 4a) Show that if 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 solve the IVP above, and if we 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 solve the first order system 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 . 4b) Conversely, show that if 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 solve the first order IVP, then 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 solve the original second order IVP.4) A special case of the discussion on the previous page is if you have a single higher order DE for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==; it is equivalent to a first order system 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. For example, consider this second order underdamped IVP for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==: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 .Exercise 5) 5a) Convert this single second order IVP into an equivalent first order system IVP for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and 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.5b) Solve the second order IVP in order to deduce a solution to the first order IVP. Use Chapter 5 methods even though you love Laplace transform more.5c) How does the Chapter 5 "characteristic polynomial" in 5b compare with the Chapter 6 (eigenvalue) "characteristic polynomial" for the first order system matrix in 5a? hmmm.5d) Is your analytic solution 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 in 5b consistent with the parametric curve shown below? (This screenshot was generated with "pplane", the sister program to "dfield" that we used in Chapters 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metric interpretation of first order systems of differential equations. (And how numerical differential equation solvers like pplane work.)Consider the general initial value problem for a first order system of differential equations: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\302\267 From either a multivariable calculus course, or from physics, recall the geometric/physical interpretation of 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 as the tangent/velocity vector to the parametric curve of points with position vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KFEieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGMi9GN1EnaXRhbGljRicvRjdRJ25vcm1hbEYnRkZGRg==, as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= varies. This picture should remind you of the discussion, but ask questions if this is new to 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the reason that the vector of derivatives 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 computed component by component is actually a limit of scaled secant vectors (and therefore a tangent/velocity vector) is: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,provided each component function is differentiable. Therefore, the reason you expect a unique solution to the IVP for a first order system is that you know where you start (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), and you know your "velocity" vector (depending on time and current location) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEqJkltcGxpZXM7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkRGLw== you expect a unique solution! (Plus, you could use something like a vector version of Euler's method or the Runge-Kutta method to approximate it!)Exercise 6) Return to the page 1 tank 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6a) Interpret the parametric solution curve 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to this IVP, as indicated in the pplane screen shot below. Notice how it follows the "velocity" vector field (which is time-independent), and how the "particle motion" location 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 is actually the vector of solute amounts in each tank. If your system involved ten coupled tanks rather than two, then this "particle" is moving around in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIzEwRidGNS9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGNQ== .6b) What are the apparent limiting solute amounts in each tank?6c) How could your smart-alec younger sibling have told you the answer to 6b without considering any differential equations at 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