Math 2250-1 Tues Oct 16 5.1-5.2 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GIzYlLUZKNiVRI3RoRidGTUZQRk1GUC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGLw== order linear differential equations, and vector space theory connections.Recap: Yesterday we started talking about second order linear differential equations for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTEYrLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRi9GMkY5RjlGOQ==: 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.The differential equation is called homogeneous. in case LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is the zero function, i.e. in 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 .Otherwise its called inhomogeneous.\302\267 For the "operator" 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 we checked the linearity properties (1) 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 (2) 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 .As a result of (just) these linearity properties we were able to understand the following two facts:Theorem 0: The solution space to the homogeneous linear 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 a subspace.Theorem 1: The general solution to the inhomogeneous second order linear 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is 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlBGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn is any single particular solution and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIkhGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn is the general solution to the homogeneous DE. (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIkhGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn is called LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImNGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn, for complementary solution, in the text).At the end of class we looked at the existence-uniqueness theorem for initial value problems:Theorem 2 (Existence-Uniqueness Theorem): Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSIsRidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJxRidGL0YyRjVGPy1GLDYlUSJmRidGL0YyRjVGPQ== be specified continuous functions on the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , and let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPS1JI21vR0YkNi1RKiZFbGVtZW50O0YnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVy1GLzYlUSJJRidGMkY1Rj4= . Then there is a unique solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSJ+RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURj0=to the initial value 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== exists and is twice continuously differentiable on the entire interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= .\302\267 Finish Monday's notes and examples to illsturate these facts, as well as the following result about the dimension of the solution space to the homogeneous DE:Theorem 3: The solution space to the second order homogeneous linear differential 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 2-dimensional.After carefully finishing Monday's notes it's not too hard to see how the discussion about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbWlHRiQ2JVEjbmRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGO0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy order DE's generalizes to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3RoRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn order DE's. We can read this portion of the notes together and see that the ideas correspond exactly to our discussion for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk= .Definition: An LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3RoRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJSp1bmRlcmxpbmVHRjQvRjZRJ25vcm1hbEYnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=order linear differential equation for a function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tRiw2I1EhRidGPQ== is a differential equation that can be written in the formLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY7LUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJ4RidGMkY1L0Y2USdub3JtYWxGJ0ZILUklbXN1cEdGJDYlLUYvNiVRInlGJ0YyRjUtRiM2JS1GQTYkRjhGSEYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R0Y/LUkjbW9HRiQ2LVEiK0YnRkgvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRmZuLyUpc3RyZXRjaHlHRmZuLyUqc3ltbWV0cmljR0Zmbi8lKGxhcmdlb3BHRmZuLyUubW92YWJsZWxpbWl0c0dGZm4vJSdhY2NlbnRHRmZuLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGZW8tRiw2JUYuLUYjNidGOi1GVzYtUSgmbWludXM7RidGSEZaRmduRmluRltvRl1vRl9vRmFvRmNvRmZvLUkjbW5HRiQ2JFEiMUYnRkhGMkY1Rj1GQC1GSzYlRk0tRiM2JS1GQTYkRmpvRkhGMkY1RlQtRlc2LVEifkYnRkhGWkZnbkZpbkZbb0Zdb0Zfb0Zhby9GZG9RJjAuMGVtRicvRmdvRl1xRlYtRlc2LVEjLi5GJ0ZIRlpGZ25GaW5GW29GXW9GX29GYW9GY29GXnEtRlc2LVEiLkYnRkhGWkZnbkZpbkZbb0Zdb0Zfb0Zhb0ZccUZecUZpcEZWLUYsNiVGLi1GIzYlRl9wRjJGNUY9RkBGTS1GVzYtUSgmcHJpbWU7RidGSEZaRmduRmluRltvRl1vRl9vRmFvL0Zkb1EsMC4xMTExMTExZW1GJ0ZecUZWLUYsNiVGLi1GIzYlLUZgcDYkRj9GSEYyRjVGPUZARk0tRlc2LVEiPUYnRkhGWkZnbkZpbkZbb0Zdb0Zfb0Zhby9GZG9RLDAuMjc3Nzc3OGVtRicvRmdvRmhyLUYvNiVRIkZGJ0YyRjVGQEZI .We search for solution functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==defined on some specified interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= of the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjxGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ4RidGL0YyRjUtRiw2JVEiYkYnRi9GMkY5, or 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 or (usually) the entire real line 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. In this chapter we assume the function 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 on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, and divide by it in order to rewrite the differential equation in the standard 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 .(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 are all functions of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, and the DE above means that equality holds for all value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= .)This DE is called linear because the operator LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= defined 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satisfies the so-called linearity properties (1) 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 (2) 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 .\302\267 The proof that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= satisfies the linearity proporties is just the same as it was for the case when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk=, that we checked yesterday. Then, since the 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 proof only depended on the linearity properties of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, just like yesterday, we deduce both of Theorems 0 and 1:Theorem 0: The solution space to the homogeneous linear 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is a subspace.Theorem 1: The general solution to the nonhomogeneous LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3RoRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn order linear 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is 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlBGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn is any single particular solution and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIkhGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn is the general solution to the homogeneous DE. (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIkhGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn is called LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImNGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn, for complementary solution, in the text).Later in the course we'll understand LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3RoRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn order existence uniqueness theorems for initial value problems, in a way analogous to how we understood the first order theorem using slope fields, but let's postpone that discussion and just record the following true theorem as a fact:Theorem 2 (Existence-Uniqueness Theorem): Let 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 be specified continuous functions on the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , and let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPS1JI21vR0YkNi1RKiZFbGVtZW50O0YnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVy1GLzYlUSJJRidGMkY1Rj4= . Then there is a unique solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSJ+RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURj0=to the initial value problemLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtRi82JVEibkYnRjJGNS9GNlEnbm9ybWFsRidGQkZCLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIitGJ0ZCLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZNLyUpc3RyZXRjaHlHRk0vJSpzeW1tZXRyaWNHRk0vJShsYXJnZW9wR0ZNLyUubW92YWJsZWxpbWl0c0dGTS8lJ2FjY2VudEdGTS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRmZuLUklbXN1YkdGJDYlLUYvNiVRImFGJ0YyRjUtRiM2JkY/LUZINi1RKCZtaW51cztGJ0ZCRktGTkZQRlJGVEZWRlhGWkZnbi1JI21uR0YkNiRRIjFGJ0ZCRkIvJS9zdWJzY3JpcHRzaGlmdEdGRi1GSDYtUSJ+RidGQkZLRk5GUEZSRlRGVkZYL0ZlblEmMC4wZW1GJy9GaG5GXnAtRiw2JUYuLUYjNiQtRjs2JEZfb0ZCRkJGREZHLUZINi1RIy4uRidGQkZLRk5GUEZSRlRGVkZYRlpGX3AtRkg2LVEiLkYnRkJGS0ZORlBGUkZURlZGWEZdcEZfcEZqb0ZHLUZqbjYlRlxvLUYjNiRGZG9GQkZob0Zqb0YuLUZINi1RKCZwcmltZTtGJ0ZCRktGTkZQRlJGVEZWRlgvRmVuUSwwLjExMTExMTFlbUYnRl9wRkctRmpuNiVGXG8tRiM2JC1GZW82JEZGRkJGQkZob0Zqb0YuLUZINi1RIj1GJ0ZCRktGTkZQRlJGVEZWRlgvRmVuUSwwLjI3Nzc3NzhlbUYnL0ZobkZfci1GLzYlUSJmRidGMkY1RkI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUklbXN1YkdGJDYlLUYsNiVRInhGJ0YvRjItRiM2JS1JI21uR0YkNiRRIjBGJy9GM1Enbm9ybWFsRidGL0YyLyUvc3Vic2NyaXB0c2hpZnRHRkVGRkZGLUkjbW9HRiQ2LVEiPUYnRkYvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlAvJSlzdHJldGNoeUdGUC8lKnN5bW1ldHJpY0dGUC8lKGxhcmdlb3BHRlAvJS5tb3ZhYmxlbGltaXRzR0ZQLyUnYWNjZW50R0ZQLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGaW4tRjs2JS1GLDYlUSJiRidGL0YyRkBGSEZGLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZwcmltZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUSJ4RidGL0YyLUYjNiUtSSNtbkdGJDYkUSIwRidGOUYvRjIvJS9zdWJzY3JpcHRzaGlmdEdGam5GOUY5LUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GYW8tRlY2JS1GLDYlUSJiRidGL0YyLUYjNiUtRmhuNiRRIjFGJ0Y5Ri9GMkZbb0Y5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZwcmltZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRidGNS1JKG1mZW5jZWRHRiQ2JC1GIzYkLUklbXN1YkdGJDYlLUYsNiVRInhGJ0YvRjItRiM2JS1JI21uR0YkNiRRIjBGJ0Y5Ri9GMi8lL3N1YnNjcmlwdHNoaWZ0R0ZqbkY5RjktRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZhby1GVjYlLUYsNiVRImJGJ0YvRjItRiM2JS1GaG42JFEiMkYnRjlGL0YyRltvRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEiOkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDRi8=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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== exists and is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= times continuously differentiable on the entire interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= .Just as for the case LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk=, the existence-uniqueness theorem lets you figure out the dimension of the solution space to homogeneous linear differential equations. The proof is conceptually the same, but messier to write down because the vectors and matrices are bigger.Theorem 3: The solution space to the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3RoRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn order homogeneous linear differential 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-dimensional.proof: By the existence half of Theorem 2, we know that there are solutions for each possible initial value problem for this (homogenenous case) of the IVP for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3RoRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn order linear DEs. So, pick solutions 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 so that their vectors of initial values (which we'll call initial value vectors)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are a basis for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 (i.e. these LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= vectors are linearly independent and span LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 . (Well, you may not know how to "pick" such solutions, but you know they exist because of the existence theorem.)Claim: In this case, the solutions 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 are a basis for the solution space. In particular, every solution to the homogeneous DE is a unique linear combination of these LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= functions and the dimension of the solution space is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= .... discussion on next page.\302\267 Check that 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 span the solution space: Consider any solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== to the DE. We can compute its vector of initial 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 .Now consider a linear combination 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 . Compute its initial value vector, and notice that you can write it as the product of the Wronskian matrix at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPUY+ times the vector of linear combination coefficients:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiktRiM2KC1JJ210YWJsZUdGJDY4LUkkbXRyR0YkNiYtSSRtdGRHRiQ2KC1GIzYqLUYsNiZRInpGJy8lJXNpemVHUSI5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRklRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZTLyUpc3RyZXRjaHlHRlMvJSpzeW1tZXRyaWNHRlMvJShsYXJnZW9wR0ZTLyUubW92YWJsZWxpbWl0c0dGUy8lJ2FjY2VudEdGUy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlxvLUYwNiUtRiM2KC1JJW1zdWJHRiQ2JS1GLDYmUSJ4RidGQkZFRkgtRiM2KC1JI21uR0YkNiVRIjBGJ0ZCRk9GQi8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lKXJlYWRvbmx5R0ZHLyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGTy8lL3N1YnNjcmlwdHNoaWZ0R0ZecEZCRl9wRmJwRmRwRk9GQkZPRkJGX3BGYnBGZHBGTy8lKXJvd2FsaWduR0YuLyUsY29sdW1uYWxpZ25HRi4vJStncm91cGFsaWduR0YuLyUocm93c3BhbkdRIjFGJy8lK2NvbHVtbnNwYW5HRmFxRmlwRltxRl1xLUY4NiYtRjs2KC1GIzYqRj8tRkw2LlEoJnByaW1lO0YnRkJGT0ZRRlRGVkZYRlpGZm5GaG4vRltvUSwwLjExMTExMTFlbUYnRl1vLUZMNi5RIihGJ0ZCRk8vRlJGR0ZUL0ZXRkdGWEZaRmZuRmhuL0Zbb1EsMC4xNjY2NjY3ZW1GJy9GXm9GZXItRmRvNiVGZm8tRiM2JkZbcEZCRkVGSEZncC1GTDYuUSIpRidGQkZPRmJyRlRGY3JGWEZaRmZuRmhuRmRyRmZyRkJGRUZIRmlwRltxRl1xRl9xRmJxRmlwRltxRl1xLUY4NiYtRjs2KC1GIzYmLUZMNi5RIjpGJ0ZCRk9GUUZURlZGWEZaRmZuRmhuL0Zbb1EsMC4yNzc3Nzc4ZW1GJy9GXm9GaHNGQkZFRkhGaXBGW3FGXXFGX3FGYnFGaXBGW3FGXXEtRjg2Ji1GOzYoLUYjNiktSShtc3Vic3VwR0YkNihGPy1GIzYmRitGQkZFRkgtRiM2KkZfci1GLDYmUSJuRidGQkZFRkgtRkw2LlEoJm1pbnVzO0YnRkJGT0ZRRlRGVkZYRlpGZm5GaG4vRltvUSwwLjIyMjIyMjJlbUYnL0Zeb0ZedS1GXHA2JUZhcUZCRk9GW3NGQkZFRkgvJTFzdXBlcnNjcmlwdHNoaWZ0R0ZecEZncC9JK21zZW1hbnRpY3NHRiRRLFtub25lLG5vbmVdRidGX3JGZ3JGW3NGQkZFRkhGaXBGW3FGXXFGX3FGYnFGaXBGW3FGXXEvJSZhbGlnbkdRJWF4aXNGJy9GanBRKWJhc2VsaW5lRicvRlxxUSdjZW50ZXJGJy9GXnFRJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0ZHLyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGZHYvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGX3cvJSZmcmFtZUdGX3cvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR0ZTLyUtZXF1YWxjb2x1bW5zR0ZTLyUtZGlzcGxheXN0eWxlR0ZTLyUlc2lkZUdRJnJpZ2h0RicvJTBtaW5sYWJlbHNwYWNpbmdHRlx3RkJGX3BGYnBGZHBGT0ZCRk8vRmV1USpDb2xWZWN0b3JGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZieC1GTDYuUSI9RidGQkZPRlFGVEZWRlhGWkZmbkZobkZnc0Zpcy1JKG1hY3Rpb25HRiQ2JC1GIzYnLUYwNiktRiM2KC1GNTY4LUY4NiktRjs2KC1GIzYqLUZkbzYlLUYsNiZRInlGJ0ZCRkVGSC1GIzYoRmB1RkJGX3BGYnBGZHBGT0ZncEZLRl9vRkJGX3BGYnBGZHBGT0ZpcEZbcUZdcUZfcUZicS1GOzYoLUYjNiotRmRvNiVGYHotRiM2KC1GXHA2JVEiMkYnRkJGT0ZCRl9wRmJwRmRwRk9GZ3BGS0Zfb0ZCRl9wRmJwRmRwRk9GaXBGW3FGXXFGX3FGYnEtRjs2KC1GIzYnLUZMNi5RIy4uRidGQkZPRlFGVEZWRlhGWkZmbkZobkZddUZdby1GTDYuUSIuRidGQkZPRlFGVEZWRlhGWkZmbkZobkZqbkZdb0ZCRkVGSEZpcEZbcUZdcUZfcUZicS1GOzYoLUYjNiotRmRvNiVGYHotRiM2KEZndEZCRl9wRmJwRmRwRk9GZ3BGS0Zfb0ZCRl9wRmJwRmRwRk9GaXBGW3FGXXFGX3FGYnFGaXBGW3FGXXEtRjg2KS1GOzYoLUYjNipGXnpGanFGX29GQkZfcEZicEZkcEZPRmlwRltxRl1xRl9xRmJxLUY7NigtRiM2KkZpekZqcUZfb0ZCRl9wRmJwRmRwRk9GaXBGW3FGXXFGX3FGYnFGYFtsLUY7NigtRiM2KkZgei1GZG82JUZqcUZgXGxGZ3BGX29GQkZfcEZicEZkcEZPRmlwRltxRl1xRl9xRmJxRmlwRltxRl1xLUY4NilGYHNGYHNGYFtsRmBzRmlwRltxRl1xLUY4NiktRjs2KC1GIzYqLUZhdDYoRmB6LUYjNigtSSVtc3VwR0YkNiVGYHVGK0ZidUZCRl9wRmJwRmRwRk9GZXRGYnVGZ3BGZHVGS0Zfb0ZCRl9wRmJwRmRwRk9GaXBGW3FGXXFGX3FGYnEtRjs2KC1GIzYqLUZhdDYoRmB6RltbbEZldEZidUZncEZkdUZLRl9vRkJGX3BGYnBGZHBGT0ZpcEZbcUZdcUZfcUZicUZgW2wtRjs2KC1GIzYpLUZhdDYoRmB6RmBcbEZldEZidUZncEZkdUZfb0ZCRl9wRmJwRmRwRk9GaXBGW3FGXXFGX3FGYnFGaXBGW3FGXXFGZ3VGanVGXHZGXnZGYHZGYnZGZXZGZ3ZGanZGXXdGYHdGYndGZHdGZ3dGaXdGW3hGXXhGYHhGQkZfcEZicEZkcEZPRkJGTy9GZXVRJ01hdHJpeEYnRmR4Rmd4Rl1fbC1GMDYpLUYjNigtRjU2OC1GODYmLUY7NigtRmRvNiUtRiw2JlEiY0YnRkJGRUZIRmN6RmdwRmlwRltxRl1xRl9xRmJxRmlwRltxRl1xLUY4NiYtRjs2KC1GZG82JUZbYGxGW1tsRmdwRmlwRltxRl1xRl9xRmJxRmlwRltxRl1xRl5zLUY4NiYtRjs2KC1GZG82JUZbYGxGYFxsRmdwRmlwRltxRl1xRl9xRmJxRmlwRltxRl1xRmd1Rmp1Rlx2Rl52RmB2RmJ2RmV2Rmd2Rmp2Rl13RmB3RmJ3RmR3Rmd3Rml3Rlt4Rl14RmB4RkJGX3BGYnBGZHBGT0ZCRk9GYnhGZHhGZ3hGYnhGQkZPRl1fbC8lK2FjdGlvbnR5cGVHUS5ydGFibGVhZGRyZXNzRidGKy1GXnk2JEYrRmpgbEYrRk8= .We've chosen the 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 so that the Wronskian matrix at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPUY+ has an inverse, so the matrix 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has a unique solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KFEiY0YnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYtUSJ+RicvRjdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlNGQA== . For this choice of linear combination coefficients, the solution 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 has the same initial value vector at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPUY+ as the solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== . By the uniqueness half of the existence-uniqueness theorem, we conclude thatLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GIzYwLUklbXN1YkdGJDYlLUYsNiVRImNGJ0YvRjItRiM2JC1JI21uR0YkNiRRIjFGJ0Y5RjkvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GUjYlRitGV0Znbi1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmBvLUZSNiVGVC1GIzYkLUZaNiRRIjJGJ0Y5RjlGZ24tRlI2JUYrRmRvRmduRlxvLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GX3AtRjY2LVEjLi5GJ0Y5RjtGPkZARkJGREZGRkhGX29GYHAtRjY2LVEiLkYnRjlGO0Y+RkBGQkZERkZGSEZecEZgcEZbcEZcby1GUjYlRlQtRiM2JC1GLDYlUSJuRidGL0YyRjlGZ24tRlI2JUYrRmlwRmduRjktRiw2I1EhRidGOQ== .Thus 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 span the solution space. If a linear combination 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, then because the zero function has zero initial vector 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 the matrix equation above implies that 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, so 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 are also linearly independent. Thus, 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 are a basis for the solution space and the general solution to the homogeneous DE can be written 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 .\342\226\241Let's do some new exercises that tie these ideas together.Exercise 1) Consider the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbWlHRiQ2JVEjcmRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGO0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy order linear homogeneous DE for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==: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.Find a basis for the 3-dimensional solution space, and the general solution. Make sure to use the Wronskian matrix (or determinant) to verify you have a basis. Hint: try exponential functions.Exercise 2a) Find the general solution to 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 .Hint: First try to find a particular solution ... try a constant function.b) Set up the linear system to solve the initial value problem for this DE, with 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 .for fun now, but maybe not just for fun later:QyYtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJKERFdG9vbHNHRiUhIiItSSdkc29sdmVHRiU2IzwmLywqLUklZGlmZkdGJjYmLUkieUdGKDYjSSJ4R0YoRjhGOEY4IiIiLUYzNiVGNUY4RjgiIiQtRjM2JEY1RjhGK0Y1ISIkIiInLy1GNjYjIiIhRisvLS1JIkRHRiU2I0Y2RkMiIiMvLS0tSSNAQEdGJTYkRkhGSkZJRkMiIihGOQ==JSFH