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Tips and Techniques:Green's Functions for Second-Order ODEsRobert J. LopezEmeritus Professor of Mathematics and Maple FellowMaplesoftLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=More than 50 years ago in a graduate course in differential equations, my colleagues and I struggled to understand what a Green's function for an ordinary differential equation really was. One wag even pointed to a green chalk mark on a blackboard and declared it to be the long-sought essence of the Green's function.It was a considerable time later that I articulated a terse description of the Green's function, namely, that it is the kernel of the integral operator that inverts a differential operator. In fact, I felt free to share this insight with students learning the Laplace transform in my undergraduate differential equations courses at Rose-Hulman Institute of Technology. The syllabi for these courses included a discussion of the Dirac delta function, from which the notions of transfer function, fundamental solution, and Green's function for the initial value problem easily follow. Of course, I was fortunate that the students in these courses had Maple at hand, either on desk-top computers or laptops. The ability to obtain solutions of test problems directly, using Maple's dsolve command, and then by a Green's function technique meant that the learning process was self-correcting. It was always possible to obtain instant feedback to determine whether or not a given approach to a computation was viable.JSFHExplaining "the kernel of the integral operator that inverts the differential operator" really isn't that hard. Use a simple example such as the initial value 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 which the solution 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JSFHJSFHIt's not hard to convince an undergraduate that the more complex initial value 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JSFHJSFHmight just have a solution of the formJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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JSFHJSFHwhere the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEidEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEidUYnRi9GMkY1RjJGREYyRkRGMkZE, the Green's function, arises from the additional complexity of the second-order differential operator.JSFHIn this month's article, we will detail how Maple might be used to obtain Green's functions for second-order initial value problems (IVPs) and boundary value problems (BVPs). For each, we examine the cases of homogeneous and nonhomogeneous conditions; for the BVP, we examine both the unmixed and mixed cases. In particular, the differential equation 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, where 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, is used in each example listed in Table 1. The initial and boundary conditions that distinguish the six examples considered, are listed in the table.
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=BVPSeparated, 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=BVPSeparated, 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BVPMixed, 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=BVPMixed, nonhomogeneousTable 1 Initial or boundary conditions delineating the six examples of this discussion
JSFHFor the IVP, the independent variable is taken as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn, whereas for the BVPs, it is taken as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn.<Text-field style="Heading 1" layout="Heading 1">Initializations</Text-field>JSFHThe following installations, definitions, and notational devices are used to make Maple's calculations more like what one might read in a textbook.
Install the inttrans package for access to the Laplace transform.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKWludHRyYW5zRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlZGPUZAAccess the Wronskian command in the VectorCalculus package.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Alias Y to the Laplace transform of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA=.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 \342\204\222 to the laplace transform command.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEmYWxpYXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSgmIzg0NjY7RidGL0YyLUkjbW9HRiQ2LlEiPUYnRi9GMi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQy8lKXN0cmV0Y2h5R0ZDLyUqc3ltbWV0cmljR0ZDLyUobGFyZ2VvcEdGQy8lLm1vdmFibGVsaW1pdHNHRkMvJSdhY2NlbnRHRkMvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZSLUYsNiVRKGxhcGxhY2VGJ0YvRjIvJStleGVjdXRhYmxlR0ZDL0YzUSdub3JtYWxGJ0ZaLUY+Ni1RIjpGJ0ZaRkFGREZGRkhGSkZMRk5GUEZTRlhGWg==Alias 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 to invlaplace, the inverse Laplace transform command.
Every instance of this symbol has to be set as an Atomic Variable.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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0 to the Dirac "delta" function.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 left-hand side of the ODE for the IVPs.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The left-hand side of the ODE for the BVPs.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEjMjBGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GVy1GLDYlUSJ5RidGL0YyLUY2Ni1RKCZQcmltZTtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTExMTExMWVtRidGWC1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORl9vLUZQNiRRIzIzRidGOUZTRlktRjY2LVEoJnByaW1lO0YnRjlGO0Y+RkBGQkZERkZGSEZpbkZYRltvLUZQNiRRIjZGJ0Y5RlNGWS1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS8lK2V4ZWN1dGFibGVHRj1GOQ==Context Menu: Assign 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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRM2Fzc2lnbn5hc35mdW5jdGlvbkYnRi9GMkY1RkU=proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );
Obtain the Green's function for the IVP 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, and use it to obtain the solution to the given IVP.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" layout="Heading 2">Mathematical Solution</Text-field><Text-field style="Heading 3" layout="Heading 3">Method 1</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Let L be the differential operator 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 and let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= be the (weak) solution of the 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Then the Green's function for the IVP 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 is 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, and the solution to this IVP is 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.Because LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= is a weak solution of the ODE LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEjTHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIj1GJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk4tRiw2JlEnJiM5NDg7RicvRjBGNEYyRjxGMkY8, it does not have a continuous first derivative. Indeed, it will later be shown that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEidEYnRi9GMkY1LUkjbW9HRiQ2LlEoJm1pbnVzO0YnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGVi1GLDYmUSJ1RidGL0YyRjVGMkZERjJGREYyRkQ= has a jump of 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 across the line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GLDYmUSJ1RidGL0YyRjVGMkY8.<Text-field style="Heading 3" layout="Heading 3">Method 2</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Let 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 be a fundamental set for the homogeneous ODE LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEjTHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIj1GJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk4tSSNtbkdGJDYlUSIwRidGMkY8RjJGPA==, and let 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 be the Wronskian for this fundamental set. Then the Green's function for the IVP LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbWlHRiQ2JlEjTHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIj1GJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk4tRiw2JlEiZkYnRi9GMkY1LUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieUYnRi9GMkY1RjJGPEYyRjwtRjk2LlEiLEYnRjJGPEY+L0ZBRjFGQkZERkZGSEZKL0ZNUSYwLjBlbUYnL0ZQUSwwLjMzMzMzMzNlbUYnRlktRlU2JS1GIzYlLUkjbW5HRiQ2JVEiMEYnRjJGPEYyRjxGMkY8RjgtSSZtb3ZlckdGJDYlRlktRiM2Ji1GOTYuUSIuRidGMkY8Rj5GQEZCRkRGRkZIRkpGam4vRlBGW29GL0YyRjUvRktGMUZeb0Y4RmJvRjJGPA== 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The astute reader will see this form of the Green's function arising from the particular solution generated by the Variation of Parameters technique. Indeed, let the general solution of the ODE LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEjTHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIj1GJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk4tRiw2JlEiZkYnRi9GMkY1LUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1RjJGPEYyRjxGMkY8 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The inert initial conditions require LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GLDYmUSJCRidGL0YyRjVGOC1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8, and the particular solution can be compressed into the single 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whence the definition of the Green's function.<Text-field style="Heading 2" layout="Heading 2">Maple's <Font italic="true">dsolve</Font> Solution</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Invoking Maple's dsolve command leads to the following solution.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Figure 1 contains a graph of this solution.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYmUSVwbG90RicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JS1GIzZGLUkmbWZyYWNHRiQ2KC1GIzYmLUkjbW5HRiQ2JVEkNTEyRidGNy9GO1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGSy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGUy8lKXN0cmV0Y2h5R0ZTLyUqc3ltbWV0cmljR0ZTLyUobGFyZ2VvcEdGUy8lLm1vdmFibGVsaW1pdHNHRlMvJSdhY2NlbnRHRlMvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zcby1JJW1zdXBHRiQ2JS1GTjYuUS8mRXhwb25lbnRpYWxFO0YnRjdGS0ZRRlRGVkZYRlpGZm5GaG5Gam4vRl5vUSwwLjExMTExMTFlbUYnLUYjNiUtRk42LlEqJnVtaW51czA7RidGN0ZLRlFGVEZWRlhGWkZmbkZobi9GW29RLDAuMjIyMjIyMmVtRicvRl5vRl1wLUZDNigtRiM2Ji1GSDYlUSIzRidGN0ZLRk0tRiw2JlEidEYnRjRGN0Y6RkstRkg2JVEiNEYnRjdGSy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYXEvJSliZXZlbGxlZEdGU0ZLLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZLLUZINiVRIjdGJ0Y3RktGXHFGX3FGYnFGZHEtRk42LlEoJm1pbnVzO0YnRjdGS0ZRRlRGVkZYRlpGZm5GaG5GXHBGXnAtRkM2KC1GIzYmLUZINiVRJTMzNzVGJ0Y3RktGTS1GYG82JUZiby1GIzYlRmlvLUZDNigtRiM2Ji1GSDYlUSIyRidGN0ZLRk1GZnBGSy1GSDYlUSI1RidGN0ZLRlxxRl9xRmJxRmRxRktGZnFGS0ZpcUZccUZfcUZicUZkcS1GTjYuUScmcGx1cztGJ0Y3RktGUUZURlZGWEZaRmZuRmhuRlxwRl5wLUZINiVRIzE4RidGN0ZLRk0tRmBvNiVGZnBGXnNGZnFGXHItRkg2JVEkMTM4RidGN0ZLLUZONi5RIn5GJ0Y3RktGUUZURlZGWEZaRmZuRmhuRmpuRl1vRmZwLUZONi5RIitGJ0Y3RktGUUZURlZGWEZaRmZuRmhuRlxwRl5wLUZINiVRJDQwOUYnRjdGSy1GTjYuUSIsRidGN0ZLRlEvRlVGNkZWRlhGWkZmbkZobkZqbi9GXm9RLDAuMzMzMzMzM2VtRidGZnAtRk42LlEiPUYnRjdGS0ZRRlRGVkZYRlpGZm5GaG4vRltvUSwwLjI3Nzc3NzhlbUYnL0Zeb0ZidS1GSDYlRmhxRjdGSy1GTjYuUSMuLkYnRjdGS0ZRRlRGVkZYRlpGZm5GaG5GXHBGXW9GXnNGaHQtRiw2JlElc2l6ZUYnRjRGN0Y6Rl51LUY+NictRiM2Ji1GSDYlUSQyMDBGJ0Y3RktGaHRGYHZGS0Y3RksvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGaHQtRiw2JlEoc2NhbGluZ0YnRjRGN0Y6Rl51LUYsNiZRLGNvbnN0cmFpbmVkRidGNEY3RjpGaHQtRiw2JlEqdGlja21hcmtzRidGNEY3RjpGXnUtRj42Jy1GIzYmRmlwRmh0RmNwRktGN0ZLRmN2RmZ2Rmh0LUYsNiZRJ2xhYmVsc0YnRjRGN0Y6Rl51LUY+NictRiM2JkZmcEZodC1GLDYmUSJ5RidGNEY3RjpGS0Y3RktGY3ZGZnZGS0Y3RktGS0YrLyUrZXhlY3V0YWJsZUdGU0ZLFigure 1 Solution of the IVP in Example 1
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" layout="Heading 2">Maple Implementation of Method 1</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=To obtain a weak solution of the equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEjTHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIj1GJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk4tRiw2JlEnJiM5NDg7RicvRjBGNEYyRjxGMkY8, the Laplace transform can be used because the ODE in Example 1 has constant coefficients. After applying the transform to the equation, use the eval command to impose the inert initial conditions. Use the solve command to obtain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEic0YnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= and note that in certain disciplines, this is usually called the transfer function, V.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Obtain 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. The inverse Laplace transform of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEic0YnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= is returned as an expression. To convert this to a Maple function, use the unapply command.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Using LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEidEYnRi9GMkY1LUkjbW9HRiQ2LlEoJm1pbnVzO0YnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGVi1GLDYmUSJ1RidGL0YyRjVGMkZERjJGREYyRkQ=, the nonzero rule in the Green's function, the solution of the IVP in Example 1 is obtained in terms of hyperbolic functions, which can be changed to exponential functions via the convert command.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=A comparison of this solution to Maple's dsolve solution, shows that the two are identical.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Figure 2 is a graph of the Green's function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEidEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEidUYnRi9GMkY1RjJGREYyRkRGMkZE, while Figure 3 is a graph of the derivative LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiZRI2RHRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Ji1GLzYmUSNkdEYnRjJGNUY4RjJGNUY4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZFLyUpYmV2ZWxsZWRHRjdGNS9GOVEnbm9ybWFsRic=. Note the continuity of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEidEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEidUYnRi9GMkY1RjJGREYyRkRGMkZE but the jump of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JVEiMUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEiL0YnRi9GMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMTY2NjY2N2VtRicvJSdyc3BhY2VHRkotRiw2JVEjMjBGJ0YvRjJGL0Yy in the derivative.
module()
local p1,v,G;
v:=t->2*exp(-23*t*(1/40))*sinh(7*t*(1/40))*(1/7);
G:=piecewise(t<u,0,t>u,v(t-u));
p1:=plot3d(G,u=0..2,t=0..2,tickmarks=[3,3,3],orientation=[10,80,0],axes=frame,axesfont=[italic,10]);
print(p1);
end module:Figure 2 Graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEidEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEidUYnRi9GMkY1RjJGREYyRkRGMkZE
module()
local p1,v,G;
v:=t->2*exp(-23*t*(1/40))*sinh(7*t*(1/40))*(1/7);
G:=piecewise(t<u,0,t>u,v(t-u));
p1:=plot3d(diff(G,t),u=0..2,t=0..2,tickmarks=[3,3,3],orientation=[65,80,0],axes=frame,axesfont=[italic,10]);
print(p1);
end module:Figure 3 Graph of t-derivative
JSFH<Text-field style="Heading 2" layout="Heading 2">Maple Implementation of Method 2</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Obtain a fundamental set for the homogeneous ODE LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LyUrZXhlY3V0YWJsZUdGPUY5 with the dsolve command.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 the unapply command to define the members of the fundamental set as the functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkE= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkE=, making each name an atomic variable.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Obtain the Wronskian of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkE= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkE=, then use the unapply command to define it as the function W.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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEidEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEidUYnRi9GMkY1RjJGREYyRkRGMkZE, the nonzero rule in the definition of the Green's function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEidEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEidUYnRi9GMkY1RjJGREYyRkRGMkZE.Note the use of the expand and combine commands, needed to compress the expression.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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEidEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEidUYnRi9GMkY1RjJGREYyRkRGMkZE to obtain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA=.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 = JSFHCompare to the solution obtained with Maple's dsolve command.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
Use a Green's function technique to solve the IVP 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. In particular, take LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSI9RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjNGJ0YyRjRGMkY0 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NDY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSI9RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjVGJ0YyRjRGMkY0.
JSFH<Text-field style="Heading 2" layout="Heading 2">Mathematical Solution</Text-field>JSFHAs in Method 2 for Example 1, the general solution of this IVP is of the form 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, where 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 is the homogeneous solution and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEicEYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRidGNS9GOVEnbm9ybWFsRic= is the particular solution obtained by Variation of Parameters.It is not hard to show that 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. Hence, A and B are determined by the two 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 and 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 obvious techniques give 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 and 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= is the Wronskian of the homogeneous solutions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkE= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkE=. The complete solution of the IVP is then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY2LUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GLDYmUSJBRidGL0YyRjUtRjk2LlEifkYnRjJGPEY+RkBGQkZERkZGSEZKL0ZNUSYwLjBlbUYnL0ZQRlgtSSVtc3ViR0YkNiVGKy1GIzYmLUkjbW5HRiQ2JVEiMUYnRjJGPEYvRjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUY5Ni5RIitGJ0YyRjxGPkZARkJGREZGRkhGSi9GTVEsMC4yMjIyMjIyZW1GJy9GUEZkby1GLDYmUSJCRidGL0YyRjVGVC1GZW42JUYrLUYjNiYtRmpuNiVRIjJGJ0YyRjxGL0YyRjVGXW9GYG8tSShtc3Vic3VwR0YkNictRjk2LlErJkludGVncmFsO0YnRjJGPC9GP1EmdW5zZXRGJy9GQUZncC9GQ0YxL0ZFRmdwL0ZHRjEvRklGZ3AvRktGZ3BGV0ZZLUYjNiUtRmpuNiVGX29GMkY8RjJGPC1GIzYlLUYsNiZRInRGJ0YvRjJGNUYyRjwvJTFzdXBlcnNjcmlwdHNoaWZ0R0Zfb0Zdby1GLDYmUSJHRidGL0YyRjUtSShtZmVuY2VkR0YkNiUtRiM2J0ZkcS1GOTYuUSIsRidGMkY8Rj4vRkFGMUZCRkRGRkZIRkpGVy9GUFEsMC4zMzMzMzMzZW1GJy1GLDYmUSJ1RidGL0YyRjVGMkY8RjJGPEZULUYsNiZRImZGJ0YvRjJGNS1GXXI2JS1GIzYlRmdyRjJGPEYyRjxGVC1GLDYmUSNkdUYnRi9GMkY1RjJGPA==, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEidEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEidUYnRi9GMkY1RjJGREYyRkRGMkZE is the Green's function for the homogeneous case detailed in Example 1.JSFH<Text-field style="Heading 2" layout="Heading 2">Maple's <Font italic="true">dsolve</Font> Solution</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Invoking Maple's dsolve command leads to the following solution when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSI9RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjNGJ0YyRjRGMkY0 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NDY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSI9RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjVGJ0YyRjRGMkY0.
module()
local q,p1;
q := 388*exp(-3*t*(1/4))*(1/7)-3230*exp(-2*t*(1/5))*(1/7)+18*t^2-138*t+409;
p1:=plot(q,t=0..3,size=[200, 200],tickmarks=[5,3],labels=[t, y]);
print(p1);
end module:Figure 4 Solution of the IVP in Example 2
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" layout="Heading 2">Maple Solution from First Principles</Text-field>JSFHThe general solution is given by 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, which evaluates toLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFH
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=where Y is used in place of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiZ0YnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y1L0Y5USdub3JtYWxGJw==, 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 is a fundamental set, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEidEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEidUYnRi9GMkY1RjJGREYyRkRGMkZE is the nonzero rule in the Green's function obtained in Example 1.Imposition of the nonhomogeneous initial conditions 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 leads to the equations
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Hence, the general solution 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 = which compares nicely to Maple's dsolve 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.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 1" layout="Heading 1">Example 3</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
For the given BVP with separated homogeneous boundary conditions, obtain an appropriate Green's function, and use it to solve the BVP.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
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" layout="Heading 2">Mathematical Solution</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=On the interval 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the boundary value 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has as its Green's 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where 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, a fundamental set for the homogeneous ODE, satisfy 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The solution of the BVP is then 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.<Text-field style="Heading 2" layout="Heading 2">Maple's <Font italic="true">dsolve</Font> Solution</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=It is convenient to define the boundary conditions as functions.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=An application of Maple's dsolve command then gives the solution as
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Figure 5 displays a graph of this solution.
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Figure 5 Solution of BVP in Example 3
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" layout="Heading 2">Maple Solution from First Principles</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The Green's function is constructed from a fundamental set 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 whose members satisfy the equations 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. To this end, take two linear combinations of the homogeneous solutions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JKG1mZW5jZWRHRiQ2JS1GIzYlLUYvNiZRInhGJ0YyRjVGOEY1RkFGNUZBRjVGQQ== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JKG1mZW5jZWRHRiQ2JS1GIzYlLUYvNiZRInhGJ0YyRjVGOEY1RkFGNUZBRjVGQQ== and impose LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkE= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkE=.
There are four constants to determine, but only two equations. Hence, solve the first equation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn in terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn; and the second, for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn in terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiLkYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTkYyRjw=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The judicious choices of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRJDE1NkYnRjJGPEYyRjw= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIzE2RidGMkY8RjJGPA== then lead to the two solutions
The Wronskian of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEjWTFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YyL0Y2USdub3JtYWxGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEjWTJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YyL0Y2USdub3JtYWxGJw==, obtained with Maple's Wronskian command, is
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The solution of the BVP, computed via this Green's function, is then
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=That this solution is equivalent to YM, Maple's dsolve solution, is verified by showing the difference between it and the Green's function solution to be zero.
For the given BVP with separated nonhomogeneous boundary conditions, obtain an appropriate Green's function, and use it to solve the BVP.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY1LUkjbW5HRiQ2JVEjMjBGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RIn5GJ0YvRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnRi8vRjNRJ2l0YWxpY0YnLUY2Ni5RKCZQcmltZTtGJ0YvRjJGOUY7Rj1GP0ZBRkNGRS9GSFEsMC4xMTExMTExZW1GJ0ZKLUY2Ni5RIitGJ0YvRjJGOUY7Rj1GP0ZBRkNGRS9GSFEsMC4yMjIyMjIyZW1GJy9GS0Zobi1GLDYlUSMyM0YnRi9GMkY1RkwtRjY2LlEoJnByaW1lO0YnRi9GMkY5RjtGPUY/RkFGQ0ZFRlhGSkZaLUYsNiVRIjZGJ0YvRjJGNUZMLUY2Ni5RIj1GJ0YvRjJGOUY7Rj1GP0ZBRkNGRS9GSFEsMC4yNzc3Nzc4ZW1GJy9GS0Znby1GLDYlUSQxMDhGJ0YvRjJGNS1JJW1zdXBHRiQ2JS1GTTYmUSJ4RidGUEYvRlMtRiM2JS1GLDYlUSIyRidGL0YyRi9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGL0YyLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGNUZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLzYmUSJ5RidGMkY1RjhGNUZBRjVGQS1JI21vR0YkNi5RKCZlcXVpdjtGJ0Y1RkEvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zcby1GPjYlUSI1RidGNUZBLUZPNi5RIn5GJ0Y1RkFGUkZURlZGWEZaRmZuRmhuL0Zbb1EmMC4wZW1GJy9GXm9GZm9GSy1GRzYlLUYjNiUtRj42JUZFRjVGQUY1RkFGNUZBLUZPNi5RKCZtaW51cztGJ0Y1RkFGUkZURlZGWEZaRmZuRmhuL0Zbb1EsMC4yMjIyMjIyZW1GJy9GXm9GYnBGYm8tRj42JVEiN0YnRjVGQUZib0ZLLUZPNi5RKCZwcmltZTtGJ0Y1RkFGUkZURlZGWEZaRmZuRmhuL0Zbb1EsMC4xMTExMTExZW1GJ0Znb0Zoby1GTzYuUSI9RidGNUZBRlJGVEZWRlhGWkZmbkZobkZqbkZdby1GPjYlUSI5RidGNUZBRjVGQQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYzLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGNUZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLzYmUSJ5RidGMkY1RjhGNUZBRjVGQS1JI21vR0YkNi5RKCZlcXVpdjtGJ0Y1RkEvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zcb0Y9LUZPNi5RIn5GJ0Y1RkFGUkZURlZGWEZaRmZuRmhuL0Zbb1EmMC4wZW1GJy9GXm9GY29GSy1GRzYlLUYjNiUtRj42JVEiMUYnRjVGQUY1RkFGNUZBLUZPNi5RIitGJ0Y1RkFGUkZURlZGWEZaRmZuRmhuL0Zbb1EsMC4yMjIyMjIyZW1GJy9GXm9GYHAtRj42JVEiM0YnRjVGQUZfb0ZLLUZPNi5RKCZwcmltZTtGJ0Y1RkFGUkZURlZGWEZaRmZuRmhuL0Zbb1EsMC4xMTExMTExZW1GJ0Zkb0Zlby1GTzYuUSI9RidGNUZBRlJGVEZWRlhGWkZmbkZobkZqbkZdby1GPjYlUSI0RidGNUZBRjVGQQ==
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" layout="Heading 2">Mathematical Solution</Text-field>JSFHIf the boundary conditions in Example 3 were the 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the solution of the associated BVP is obtained by adding to the Green's function solution of Example 3, the 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JSFHRecall that 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. The solution of the BVP is then 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JSFHJSFH<Text-field style="Heading 2" layout="Heading 2">Maple's <Font italic="true">dsolve</Font> Solution</Text-field>JSFHAs in Example 3, define the boundary conditions as functions.
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 6 Solution of BVP in Example 4
JSFH<Text-field style="Heading 2" layout="Heading 2">Maple Solution from First Principles</Text-field>JSFHUsing the Green's function and the fundamental set from Example 3, the solution for Example 4 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or JSFH
JSFHThat this solution is equivalent to Maple's dsolve solution is verified by following calculation showing the difference between these two solutions is zero.JSFH
For the given BVP with mixed homogeneous boundary conditions, obtain an appropriate Green's function, and use it to solve the BVP.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
JSFH<Text-field style="Heading 2" layout="Heading 2">Mathematical Solution</Text-field>JSFHFor the second-order ODE, the general mixed boundary 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include the periodic boundary 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JSFHWe will examine two methods of solution, the first, a distributional approach in which the equation 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 is solved by the Laplace transform; and a second, gleaned from Ince's classic ODE text.JSFH<Text-field style="Heading 3" layout="Heading 3">Method 1</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Take the Laplace transform of the equation 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 and, since the initial consitions are not known, set 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. Solve the resulting equation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM4NDY2O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieUYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA=, the Laplace transform of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA=, and impose the boundary conditions 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, giving two equations in the two unknowns a and b. This determines the weak solution 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.The Green's function determined by this method is 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, where
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=and the solution of the BVP, determined by the integral 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, is
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 7 Solution of the BVP in Example 5
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 3" layout="Heading 3">Method 2</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Let 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 be a fundamental set for the homogeneous ODE LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEjTHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIj1GJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk4tSSNtbkdGJDYlUSIwRidGMkY8RjJGPA==. Form a new fundamental set 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiTEYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS9GOVEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiUkYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS9GOVEnbm9ybWFsRic= are two distinct linear combinations of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkE= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkE=, and L and R respectively represent "left" and "right". In particular, takeLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiTEYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYuUSI9RidGNS9GOVEnbm9ybWFsRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZZLUYvNiZRImFGJ0YyRjVGOC1GRDYuUSJ+RidGNUZHRklGS0ZNRk9GUUZTRlUvRlhRJjAuMGVtRicvRmVuRl1vLUYsNiUtRi82JlEieUYnRjJGNUY4LUYjNiYtSSNtbkdGJDYlUSIxRidGNUZHRjJGNUY4RkAtRkQ2LlEiK0YnRjVGR0ZJRktGTUZPRlFGU0ZVL0ZYUSwwLjIyMjIyMjJlbUYnL0ZlbkZecC1GLzYmUSJiRidGMkY1RjhGaW4tRiw2JUZhby1GIzYmLUZnbzYlUSIyRidGNUZHRjJGNUY4RkBGNUZH and 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and build the Green's function from the piecewise 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The four constants LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEiYkYnRi9GMkY1RjgtRiw2JlEiY0YnRi9GMkY1RjgtRiw2JlEiZEYnRi9GMkY1RjJGPA== are determined by four conditions: across the line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GLDYmUSJ1RidGL0YyRjVGMkY8, the continuity of G and the jump of 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 in the derivative of G, and the two boundary conditions. In particular, the continuity condition is expressed via the equation 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 while the jump condition for the derivative is expressed via the equation 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. When applying the boundary conditions, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiTEYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS9GOVEnbm9ybWFsRic= is used when evaluating at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8; and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiUkYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS9GOVEnbm9ybWFsRic=, at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjFGJ0YyRjxGMkY8.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" layout="Heading 2">Maple's <Font italic="true">dsolve</Font> Solution</Text-field>JSFHFor compactness, first define the two boundary conditions, then apply the dsolve command. The solution is returned in a form that will allow comparisons to the results obtained by other methods.
JSFH<Text-field style="Heading 2" layout="Heading 2">Maple Implementation of Method 1</Text-field>JSFHThe solution of 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 is obtained via the Laplace transform, using the unknowns a and b for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSNtbkdGJDYlUSIwRidGMi9GNlEnbm9ybWFsRidGMkZBRjJGQUYyRkE= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEoJnByaW1lO0YnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiUtRiM2JS1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8RjJGPEYyRjw=, respectively. To facilitate imposing the boundary conditions that then determine the constants a and b, the solution is written as a pair of functions YL and YR, the first applicable when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GLDYmUSJ1RidGL0YyRjVGMkY8, the second, when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPkYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GLDYmUSJ1RidGL0YyRjVGMkY8.
Obtain the Laplace transform of the ODE 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Apply the initial 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 for the unknown Laplace transform and invertLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZKLUkjbWlHRiQ2JlEldGVtcEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSomY29sb25lcTtGJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLUYsNiZRKWludHRyYW5zRidGL0YyRjUtRjk2LlEjOi1GJ0YyRjxGPkZBRkNGRUZHRklGSy9GTlEmMC4wZW1GJy9GUUZZLUYsNiZRK2ludmxhcGxhY2VGJ0YvRjJGNS1JKG1mZW5jZWRHRiQ2JS1GIzYpLUYsNiZRJnNvbHZlRidGL0YyRjUtRmluNiUtRiM2KC1GLDYjUSFGJy1JKG1hY3Rpb25HRiQ2JS1JKm12ZXJiYXRpbUdGJDYjUWpebC8sLiooIiM/IiIiKUkic0c2IiIiIyIiIi1JJyZMc2NyO0c2IjYlLUkieUc2IjYjSSJ4RzYiSSJ4RzYiSSJzRzYiIiIiIiIiKiYiIz8iIiJJImJHNiIiIiIhIiIqKCIjPyIiIkkic0c2IiIiIkkiYUc2IiIiIiEiIiooIiNCIiIiSSJzRzYiIiIiLUknJkxzY3I7RzYiNiUtSSJ5RzYiNiNJInhHNiJJInhHNiJJInNHNiIiIiIiIiIqJiIjQiIiIkkiYUc2IiIiIiEiIiomIiInIiIiLUknJkxzY3I7RzYiNiUtSSJ5RzYiNiNJInhHNiJJInhHNiJJInNHNiIiIiIiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiZJInNHNiIiIiJJInVHNiIiIiIhIiJGJy8lK2FjdGlvbnR5cGVHUTttYXBsZXNvZnQuY29tOmxhYmVsKEwxNTc0KUYnLyUldmlld0dRJmxhYmVsRictRjk2LlEiLEYnRjJGPEY+L0ZCRjFGQ0ZFRkdGSUZLRlgvRlFRLDAuMzMzMzMzM2VtRictRiw2JlEtJkxhcGxhY2V0cmY7RidGL0YyRjUtRmluNiUtRiM2KS1GLDYmUSJ5RidGL0YyRjUtRmluNiUtRiM2JC1GLDYmUSJ4RidGL0YyRjVGPEYyRjxGZHBGaHFGZHAtRiw2JlEic0YnRi9GMkY1RjxGMkY8RjxGMkY8RmRwRltyRmRwRmhxRjxGMkY8LUY5Ni5RIn5GJ0YyRjxGPkZBRkNGRUZHRklGS0ZYRlotRjk2LlEpYXNzdW1pbmdGJ0YyRjxGPkZBRkNGRUZHRklGS0ZYRlpGXnItRiw2JlEidUYnRi9GMkY1LUY5Ni5RIj5GJ0YyRjxGPkZBRkNGRUZHRklGS0ZNRlAtSSNtbkdGJDYlUSIwRidGMkY8LUY5Ni5RIjpGJ0YyRjxGPkZBRkNGRUZHRklGS0ZNRlAtSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGWS8lJmRlcHRoR0Zmcy8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GLDYmUSZ0ZW1wTEYnRi9GMkY1RjgtRiw2JlEpc2ltcGxpZnlGJ0YvRjJGNS1GaW42JS1GIzYkRitGPEYyRjxGXnJGYXJGXnJGaHEtRjk2LlEiPEYnRjJGPEY+RkFGQ0ZFRkdGSUZLRk1GUEZkckZec0Zhcy1GLDYmUSZ0ZW1wUkYnRi9GMkY1RjhGYXRGZHRGXnJGYXJGXnJGaHFGZ3JGZHJGXnNGXnIvJStleGVjdXRhYmxlR0ZARjw=JSFHDefine the two solutions as 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjWVJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJ4RidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZA =
JSFHApply the boundary conditions and solve the resulting equations for a and b. These are large expressions in which hyperbolic and exponential functions appear. Considerable simplification is obtained by converting all expressions to exponential form.
JSFHForm the Green's function and use it to calculate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA=, the solution of the BVP. The proof that all results are valid is obtained by showing the solution found with the Green's function is the same as that found by Maple's dsolve command.
Form the Green's 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Use the Green's function to obtain the solution of the 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Show that the solution via the Green's function matches Maple's dsolve 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 =
JSFH<Text-field style="Heading 2" layout="Heading 2">Maple Implementation of Method 2</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Recalling that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjRlNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic= = was a fundamental set for the IVP in Example 1, writeLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Write and solve the four equations detailed in Method 2.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
The continuity 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The jump condition for the 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 two boundary 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 the four 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
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Form the functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiTEYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS9GOVEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiUkYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS9GOVEnbm9ybWFsRic=, and from these, the piecewise function G.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
The function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiTEYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS9GOVEnbm9ybWFsRic=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The function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiUkYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS9GOVEnbm9ybWFsRic=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The Green's function 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Verify that the Green's function via Method 2 is the same as the one via Method 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
For the given BVP with mixed nonhomogeneous boundary conditions, obtain an appropriate Green's function, and use it to solve the BVP.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY1LUkjbW5HRiQ2JVEjMjBGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RIn5GJ0YvRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnRi8vRjNRJ2l0YWxpY0YnLUY2Ni5RKCZQcmltZTtGJ0YvRjJGOUY7Rj1GP0ZBRkNGRS9GSFEsMC4xMTExMTExZW1GJ0ZKLUY2Ni5RIitGJ0YvRjJGOUY7Rj1GP0ZBRkNGRS9GSFEsMC4yMjIyMjIyZW1GJy9GS0Zobi1GLDYlUSMyM0YnRi9GMkY1RkwtRjY2LlEoJnByaW1lO0YnRi9GMkY5RjtGPUY/RkFGQ0ZFRlhGSkZaLUYsNiVRIjZGJ0YvRjJGNUZMLUY2Ni5RIj1GJ0YvRjJGOUY7Rj1GP0ZBRkNGRS9GSFEsMC4yNzc3Nzc4ZW1GJy9GS0Znby1GLDYlUSQxMDhGJ0YvRjJGNS1JJW1zdXBHRiQ2JS1GTTYmUSJ4RidGUEYvRlMtRiM2JS1GLDYlUSIyRidGL0YyRi9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGL0YyLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY+LUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGNUZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLzYmUSJ5RidGMkY1RjhGNUZBRjVGQS1JI21vR0YkNi5RKCZlcXVpdjtGJ0Y1RkEvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zcby1GPjYlUSIzRidGNUZBLUZPNi5RIn5GJ0Y1RkFGUkZURlZGWEZaRmZuRmhuL0Zbb1EmMC4wZW1GJy9GXm9GZm9GSy1GRzYlLUYjNiUtRj42JUZFRjVGQUY1RkFGNUZBLUZPNi5RKCZtaW51cztGJ0Y1RkFGUkZURlZGWEZaRmZuRmhuL0Zbb1EsMC4yMjIyMjIyZW1GJy9GXm9GYnAtRj42JVEiNEYnRjVGQUZib0ZLLUZPNi5RKCZwcmltZTtGJ0Y1RkFGUkZURlZGWEZaRmZuRmhuL0Zbb1EsMC4xMTExMTExZW1GJ0Znb0Zoby1GTzYuUSIrRidGNUZBRlJGVEZWRlhGWkZmbkZobkZhcEZjcC1GPjYlUSI1RidGNUZBRmJvRkstRkc2JUY7RjVGQUZccS1GPjYlUSI3RidGNUZBRmJvRktGZ3BGYnEtRk82LlEiPUYnRjVGQUZSRlRGVkZYRlpGZm5GaG5Gam5GXW8tRj42JVEiOUYnRjVGQUY1RkE=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY/LUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGNUZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLzYmUSJ5RidGMkY1RjhGNUZBRjVGQS1JI21vR0YkNi5RKCZlcXVpdjtGJ0Y1RkEvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zcby1GPjYlUSI3RidGNUZBLUZPNi5RIn5GJ0Y1RkFGUkZURlZGWEZaRmZuRmhuL0Zbb1EmMC4wZW1GJy9GXm9GZm9GSy1GRzYlLUYjNiUtRj42JUZFRjVGQUY1RkFGNUZBLUZPNi5RIitGJ0Y1RkFGUkZURlZGWEZaRmZuRmhuL0Zbb1EsMC4yMjIyMjIyZW1GJy9GXm9GYnBGPUZib0ZLLUZPNi5RKCZwcmltZTtGJ0Y1RkFGUkZURlZGWEZaRmZuRmhuL0Zbb1EsMC4xMTExMTExZW1GJ0Znb0Zoby1GTzYuUSgmbWludXM7RidGNUZBRlJGVEZWRlhGWkZmbkZobkZhcEZjcC1GPjYlUSIzRidGNUZBRmJvRkstRkc2JS1GIzYlLUY+NiVRIjFGJ0Y1RkFGNUZBRjVGQUZpcC1GPjYlUSI4RidGNUZBRmJvRktGZHBGX3EtRk82LlEiPUYnRjVGQUZSRlRGVkZYRlpGZm5GaG5Gam5GXW8tRj42JVEiNEYnRjVGQUZib0Y1RkE=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" layout="Heading 2">Mathematical Solution</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Let 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 be a fundamental set satisfying 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. Then, to the solution in Example 5 add the 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Recalling Method 2 in Example 5, the reader should note 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 only because the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn's are functions of just LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn whereas LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiTEYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS9GOVEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiUkYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS9GOVEnbm9ybWFsRic= are functions of both LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn, but also because 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In the solution of Example 5 by Method 2, the boundary conditions were applied to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEieEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEidUYnRi9GMkY1RjJGREYyRkRGMkZE, not separately to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiTEYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS9GOVEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiUkYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS9GOVEnbm9ybWFsRic=.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" layout="Heading 2">Maple's <Font italic="true">dsolve</Font> Solution</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=For compactness, first define the two boundary conditions, then apply the dsolve command. The solution is returned in a form that will allow comparisons to the results obtained by other methods.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Figure 8 compares the graphs of the solutions from Examples 5 and 6, verifying that these two solutions are different.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
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Figure 8 The solutions for Examples 5 and 6
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" layout="Heading 2">Maple Solution from First Principles</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=To obtain the functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkE= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkE=, recall that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjRlNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic= = was a fundamental set for the IVP in Example 1, and writeJSFH
There are four constants and only two equations, so solve the first equation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn and the second for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn, obtaining
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=With judicious choices for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn, obtain the fundamental set whose members are
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Legal Notice: \302\251 Maplesoft, a division of Waterloo Maple Inc. 2017. Maplesoft and Maple are trademarks of Waterloo Maple Inc. This application may contain errors and Maplesoft is not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact Maplesoft for permission if you wish to use this application in for-profit activities.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=