Math 2250-1 Fri Oct 26 5.5 Finding LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlBGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn for non-homogeneous linear differential equations.On Wednesday we started talking about the method of undetermined coefficients for finding particular solutions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlBGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn 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i.e.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInlGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUYsNiVRImZGJ0YvRjJGPQ==(as a step in finding the general solution 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 to solve initial value problems). The idea, which we haven't made precise yet, is to pick a guess for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlBGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn that is a linear combination of functions (with undetermined linear combination coefficients) that "looks like" LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, plug the guess into the DE, and solve for the undetermined coefficients. \302\267 Finish Wednesday's notes. We're on Exercise 5.\302\267 The following Theorem justifies the method of undetermined coefficients, in both the "base case" and the more subtle case of when terms on the right side of the DE solve the homogeneous DE:Theorem: \302\267 Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= be vector spaces with the same dimension LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjxGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmaW5maW47RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlNGOQ==. \302\267 Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJWRidGL0YyLUY2Ni1RKCZzcmFycjtGJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GVi1GLDYlUSJXRidGL0YyRjk= be a linear transformation, (i.e. 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 and 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 holds 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZFbGVtZW50O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRKCYjODQ3NztGJy9GMEY9RjlGOQ==.) \302\267 Then the equation problem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNihRInZGJy8lJWJvbGRHRjFGLy8lKnVuZGVybGluZUdGMS9GM1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjNRJ25vcm1hbEYnRkYtSSNtb0dGJDYvUSI9RidGPS9GM0ZFRkMvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRk8vJSlzdHJldGNoeUdGTy8lKnN5bW1ldHJpY0dGTy8lKGxhcmdlb3BHRk8vJS5tb3ZhYmxlbGltaXRzR0ZPLyUnYWNjZW50R0ZPLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGaG4tRiw2KFEiZkYnRj1GL0Y/RkFGQ0ZG has a unique solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSomRWxlbWVudDtGJy9GN1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSJWRidGMi9GN1EnaXRhbGljRidGQA== for each 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 if and only if the only solution to 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 is 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proof:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEqJkltcGxpZXM7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkRGLw==: Assume that the equation 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 has a unique solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSomRWxlbWVudDtGJy9GN1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSJWRidGMi9GN1EnaXRhbGljRidGQA== for each 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. Thus in the special case that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2KFEiZkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSI9RicvRjdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYnUSIwRidGL0Y0L0Y3RjtGOUZA the solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSI9RicvRjdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYnUSIwRidGL0Y0L0Y3RjtGOS1GLDYjUSFGJ0ZA will be the only solution. (For linear LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= it is always true that 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, since e.g. 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 .)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRitGKy1GLDYtUTImRG91YmxlTGVmdEFycm93O0YnRi9GMkY1L0Y4USV0cnVlRidGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZMLUYsNi1RIjpGJ0YvRjJGNUY3RjlGO0Y9Rj9GS0ZNRi8= Assume that the only solution to 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 is 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step 1: Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lKnVuZGVybGluZUdGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JS1JI21uR0YkNiRRIjFGJy9GOlEnbm9ybWFsRidGNS9GOlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIixGJ0ZFLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRlIvJSpzeW1tZXRyaWNHRlIvJShsYXJnZW9wR0ZSLyUubW92YWJsZWxpbWl0c0dGUi8lJ2FjY2VudEdGUi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVGLi1GIzYlLUZCNiRRIjJGJ0ZFRjVGR0ZJRkwtRk02LVEifkYnRkVGUC9GVEZSRlVGV0ZZRmVuRmduRmluL0Zdb0Zbby1GTTYtUSMuLkYnRkVGUEZpb0ZVRldGWUZlbkZnbi9Gam5RLDAuMjIyMjIyMmVtRidGam8tRk02LVEiLkYnRkVGUEZpb0ZVRldGWUZlbkZnbkZpbkZqb0Zmby1GLDYlRi4tRiM2JS1GLzYlUSJuRidGNUZHRjVGR0ZJLUYvNiNRIUYnRkU= be a basis for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The fact that 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 are linearly independent (in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) and our hypothesis about homogeneous solutions implies that that 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 are linearly independent in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: 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then because LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5is linear it is also true 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Since the only solution to the homogeneous problem 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 is 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 it is also true 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 .step 2: Since the dimension of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is also LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, the linearly independent vectors 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 automatically span LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, and are a basis. Thus each 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 can be written as 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so the equation 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 has a solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSIuRicvRjdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlNGQA==step 3: If there is another solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEidUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjdRJ25vcm1hbEYn, i.e. 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 , then 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 . Since we're assuming the only solution to the homogeneous equation is the zero vector, we deduce 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, i.e. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2KFEidUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSI9RicvRjdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiw2KFEidkYnRi9GMkY0RjZGOUZA , and the equation 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 has exactly one solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2KFEidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSomRWxlbWVudDtGJy9GN1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSJWRidGMi9GN1EnaXRhbGljRidGQA== .\342\226\241Exercise 1) Understand the result above, and what it has to do with the method of undetermined coefficients in both the "base case" and the 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" case.Variation of Parameters: This is an alternate method for finding particular solutions. Its advantage is that is always provides a particular solution, even for non-homogeneous problems in which the right-hand side doesn't fit into a nice finite dimensional subspace preserved by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The formula for the particular solutions can be somewhat messy to work with, however, once you start computing.Here's the formula: Let 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 be a basis of solutions to the homogeneous 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 .Then 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 is a particular solution toLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInlGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUYsNiVRImZGJ0YvRjJGPQ==provided the coefficient functions (aka "varying parameters") 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 have derivatives satisfying the matrix 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where 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 is the Wronskian matrix.Here's how to check this fact when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk=: 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 .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 .SetLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYkLUYjNiotSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ1RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIxRicvRjtRJ25vcm1hbEYnRkMvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RKCZwcmltZTtGJ0ZDLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZOLyUpc3RyZXRjaHlHRk4vJSpzeW1tZXRyaWNHRk4vJShsYXJnZW9wR0ZOLyUubW92YWJsZWxpbWl0c0dGTi8lJ2FjY2VudEdGTi8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHUSYwLjBlbUYnLUYxNiUtRjQ2JVEieUYnRjdGOkY9RkUtRkk2LVEiK0YnRkNGTEZPRlFGU0ZVRldGWS9GZm5RLDAuMjIyMjIyMmVtRicvRmluRmRvLUYxNiVGMy1GIzYkLUZANiRRIjJGJ0ZDRkNGRUZILUYxNiVGXW9GaG9GRUZDRkMtRkk2LVEiPUYnRkNGTEZPRlFGU0ZVRldGWS9GZm5RLDAuMjc3Nzc3OGVtRicvRmluRmNwLUZANiRGR0ZDRkM=.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 .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 .Notice that the two LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiUtSSNtb0dGJDYtUSMuLkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR1EmMC4wZW1GJy1GMTYtUSIuRidGNEY3RjpGPEY+RkBGQkZEL0ZHRktGSUY0RjRGNA== equations are equivalent to the matrix 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which is equivalent to the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk= version of the claimed condition for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRInBGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn. Under these conditions we 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\342\226\241Exercise 2) Use variation of parameters to re-find a particular solution to 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.(Recall, a basis for the homogeneous solution space is 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 . When we used the method of undetermined coefficients we found a 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.)