Math 2250-1Mon Nov 197.1-7.2 Systems of differential equations - to model multi-component systems. Reduction of higher order IVPs to first order system IVPs. Theory and computation for first order IVPs.\302\267 Recall from Friday's notes, that IVPs for higher order differential equations or systems of differential equations are equivalent to IVPs for first order systems of DEs.\302\267 We were working an example of this equivalence, Exercise 5 , which is Exercise 1 below.We've already completed parts a,b:Consider this second order underdamped IVP for 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.Exercise 1) 1a) Convert this single second order IVP into an equivalent first order system IVP for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and 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.We got: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=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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.We got:The characteristic polynomial for the homogeneous 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 which has roots 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. Thus the general solutionto the homogeneous DE 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and matching 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 to the initial conditions 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 yields the IVP 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 .Finish 1b, to find the solution to the equivalent IVP for 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 in 1a:1c) How does the Chapter 5 "characteristic polynomial" in 1b compare with the Chapter 6 (eigenvalue) "characteristic polynomial" for the first order system matrix in 1a? Discuss.1d) Is your analytic solution 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 in 1b consistent with the parametric curve shown below? (This screenshot was generated with "pplane", the sister program to "dfield" that we used in Chapters 1-2.) Interpret the image of this parametric curve in terms of an underdamped 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After Exercise 1, finish pages 5-6 of Friday's notes.Here are the Theorems for first order systems of differential equations. They are analogous to the ones we discussed for first order scalar DE IVPs back in Chapter 1.Theorem 1 For the 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KFEiRkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSJ0RidGMi9GN1EnaXRhbGljRictSSNtb0dGJDYtUSIsRicvRjdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRk4vJSpzeW1tZXRyaWNHRk4vJShsYXJnZW9wR0ZOLyUubW92YWJsZWxpbWl0c0dGTi8lJ2FjY2VudEdGTi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNihRInhGJ0YvRjJGNEY2RjlGSkZKRko= is continuous in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZ1bWludXMwO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZMRjk=variable and differentiable in its LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjdRJ25vcm1hbEYn variable, then there exists a unique solution to the IVP, at least on some (possibly short) time interval 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 .Theorem 2 For the special case of the first order linear system of differential equations 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 the matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and the vector function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KFEiZkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGMi9GN1EnaXRhbGljRicvRjdRJ25vcm1hbEYnRkZGRg== are continuous on an open interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= containing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPUY+ then a solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KFEieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGMi9GN1EnaXRhbGljRicvRjdRJ25vcm1hbEYnRkZGRg== exists and is unique, on the entire interval.Remark: The solutions to these systems of DE's may be approximated numerically using vectorized versions of Euler's method and the Runge Kutta method. The ideas are exactly the same as they were for scalar equations, except that they now use vectors. This is how commerical numerical DE solvers work. For example, with time-step LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= the Euler loop would increment as follows: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 .Remark: These theorems are the true explanation for why the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3RoRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn-order linear DE IVPs in Chapter 5 always have unique solutions - because each LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3RoRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSgmbWludXM7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlctRi82JVEmb3JkZXJGJ0YyRjVGRA== linear DE IVP is equivalent to an IVP for a first order system of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= linear DE's. In fact, when software finds numerical approximations for solutions to higher order (linear or non-linear) DE IVPs that can't be found by the techniques of Chapter 5 or other mathematical formulas, it works by converting these IVPs to the equivalent first order system IVPs, and uses algorithms like Euler and Runge-Kutta to approximate the solutions.Theorem 3) Vector space theory for first order systems of linear DEs (Notice the familiar themes...we can completely understand these facts if we take the intuitively reasonable existence-uniqueness Theorem 2 as fact.)3.1) For vector functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KFEieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGMi9GN1EnaXRhbGljRicvRjdRJ25vcm1hbEYnRkZGRg== differentiable on an interval, the 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is linear, i.e.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 .check!3.2) Thus, by the fundamental theorem for linear transformations, the general solution to the non-homogeneous linear 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEpJkZvckFsbDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRictSSNtaUdGJDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYsNi1RKiZFbGVtZW50O0YnRi9GMkY1RjdGOUY7Rj1GPy9GQkZGRkQtRkg2JVEiSUYnRktGTkYv 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JJW1zdWJHRiQ2JS1GLDYoUSJ4RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR0Y5LyUqdW5kZXJsaW5lR0Y5LyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GIzYkLUYsNiVRInBGJ0Y6L0Y/USdpdGFsaWNGJy9GP1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0Y6RklGS0ZLRktGK0ZL is any single particular solution and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2KFEieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNy8lKnVuZGVybGluZUdGNy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JC1GLDYlUSJIRidGOC9GPVEnaXRhbGljRicvRj1RJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGOEZHRklGSUZJ is the general solution to the homogeneous problemLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2KFEieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSgmcHJpbWU7RicvRjdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHUSYwLjBlbUYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEidEYnRjIvRjdRJ2l0YWxpY0YnRkBGQC1GPTYtUSgmbWludXM7RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjIyMjIyMjJlbUYnL0ZVRl9vLUYsNiVRIkFGJ0YyRmluRldGK0ZXLUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGaG8tSSNtbkdGJDYnUSIwRidGL0Y0L0Y3RjtGOS1GLDYjUSFGJ0ZAWe frequently write this homogeneous linear system of DE's 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 .3.3) For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1zdWJHRiQ2JS1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRkAtRiM2Jy1GLDYlUSJuRidGL0YyLUkjbW9HRiQ2LVEoJnRpbWVzO0YnRkAvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRk0vJSlzdHJldGNoeUdGTS8lKnN5bW1ldHJpY0dGTS8lKGxhcmdlb3BHRk0vJS5tb3ZhYmxlbGltaXRzR0ZNLyUnYWNjZW50R0ZNLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGZm5GREYvRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLDYjUSFGJ0ZA and 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 the solution space on the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZ1bWludXMwO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZMRjk=interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to the homogeneous 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is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EmZmFsc2VGJy8lKnVuZGVybGluZUdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY1-dimensional. Here's why:\302\267 Let 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 be any LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= solutions to the homogeneous problem chosen so that the Wronskian matrix at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPS1JI21vR0YkNi1RKiZFbGVtZW50O0YnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVy1GLzYlUSJJRidGMkY1Rj4=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 invertible. (By the existence theorem we can choose solutions for any collection of initial vectors - so for example, in theory we could pick the matrix above to actually equal the identity matrix. In practice we'll be happy with any invertible matrix. )\302\267 Then for any 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 the 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 solution 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 where the linear combination coefficients are the solution to the Wronskian matrix 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 .Thus, because the Wronskian matrix at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPUY+ is invertible, every IVP can be solved with a linear combination of 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 , and since each IVP has only one solution, 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 span the solution space. The same matrix equation shows that the only linear combination that yields the zero function (which has initial vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2KFEiYkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSI9RicvRjdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYnUSIwRidGL0Y0L0Y3RjtGOUZA ) is the one with 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 Thus 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 are also linearly independent. Therefore they are a basis for the solution space, and their number LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the dimension of the solution space.7.3 Eigenvector method for finding the general solution to the homogeneous constant matrix first order system of differential 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Here's how: We look for a basis of solutions 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 , where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjdRJ25vcm1hbEYn is a constant vector. Substituting this form of potential solution into the system of DE's above yields the 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 .Dividing both sides of this equation by the scalar function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRJyYjOTU1O0YnL0YzUSZmYWxzZUYnL0Y2USdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0Y/LyUmZmVuY2VHRj4vJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS1GLzYlUSJ0RidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGPw== gives the conditionLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JJW1zdXBHRiQ2JS1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GIzYkLUYsNiNRIUYnRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYsNihRInZGJy8lJWJvbGRHUSV0cnVlRicvRjBGZm4vJSp1bmRlcmxpbmVHRmZuL0YzUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GOTYtUSI9RidGMkY8Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORmNvLUYsNiVRIkFGJ0Znbi9GM1EnaXRhbGljRidGOEZXRjI= .\302\267 We get a solution every time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjdRJ25vcm1hbEYn is an eigenvector of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= with eigenvalue LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy !\302\267 If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is diagonalizable then there is an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 basis of eigenvectors 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 and 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 are a basis for the solution space on the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSgmIzg0Nzc7RicvRjBGPUY5Rjk= , because the Wronskian matrix at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= is the invertible diagonalizing matrixLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSJbRidGOS9GPEYxRj4vRkFGMUZCRkRGRkZIL0ZLUSwwLjE2NjY2NjdlbUYnL0ZORlUtSSVtc3ViR0YkNiUtRiw2KFEidkYnLyUlYm9sZEdGMUYvLyUqdW5kZXJsaW5lR0YxL0YzUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnRjlGOS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYmLUYjNiQtRlg2JUZaLUYjNiQtRmNvNiRRIjJGJ0Y5RjlGZm9GOUY5LyUlb3BlbkdRInxnckYnLyUmY2xvc2VHRmdwLUY2Ni1RIy4uRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZOUSYwLjBlbUYnLUY2Ni1RIi5GJ0Y5RjtGPkZARkJGREZGRkgvRktGYHFGX3EtRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZkcUZfcS1GNjYtRmdwRjlGO0Y+RlNGQkZERkZGSC9GS1EsMC4xMTExMTExZW1GJy9GTkZbci1GWDYlRlotRiM2JC1GLDYlUSJuRidGL0YyRjlGZm8tRjY2LVEiXUYnRjlGUkY+RlNGQkZERkZGSEZURlxyRjk=that we considered in Chapter 6.\302\267 If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= has complex number eigenvalues and eigenvectors it may still be diagonalizable over LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDUwO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1, and we will still be able to extract a basis of real vector function solutions. If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is not diagonalizable over LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 or over LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDUwO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 the situation is more complicated. Exercise 2a) Use the method above to find the general homogeneous solution to 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 .b) Solve the IVP 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 .c) There is an equivalent second order linear differential equation IVP associated to this first order system. Are your answers above consistent with it?JSFH