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Mathematical Physics

By Carlos A. Smith

IntroductionAn Introductory ExampleModelingDifferential EquationsForcing FunctionsBook ObjectivesObjects in a Gravitational FieldAn instance Antidifferentiation: method for fixing First-Order traditional Differential EquationsBack to part 2-1Another ExampleSeparation of Variables: process for fixing First-Order traditional Differential Equations again to part 2-5Equations, Unknowns, and levels of Read more...

summary: IntroductionAn Introductory ExampleModelingDifferential EquationsForcing FunctionsBook ObjectivesObjects in a Gravitational FieldAn instance Antidifferentiation: process for fixing First-Order traditional Differential EquationsBack to part 2-1Another ExampleSeparation of Variables: procedure for fixing First-Order traditional Differential Equations again to part 2-5Equations, Unknowns, and levels of FreedomClassical recommendations of normal Linear Differential EquationsExamples of Differential EquationsDefinition of a Linear Differential EquationIntegrating issue MethodCharacteristic Equation

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Extra resources for A First Course in Differential Equations, Modeling, and Simulation

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The first term of the solution is y1 = C1 ert. 6. For case 3 (two complex roots) the roots are at r1 = α + iβ and r2 = α – iβ, where α = a1/2a2 and β = 4 a2 a0 − a12 , then 2 a2 y = C1′ e(α + iβ )t + C2′ e(α − iβ )t y = eαt C1′ e iβt + C2′ e − iβt  It is rather difficult to obtain a good qualitative indication of this response because of the complex exponential powers. A better expression, avoiding complex numbers, can be obtained using Euler’s identity eiβt = cos βt + isin βt. 7. 21 Using Euler’s identity, y = eαt C1′ e iβt + C2′ e − iβt  C1′ e iβt + C2′ e − iβt = C1′ cos β t + iC1′ sin β t + C2′ cos β t − iC2′ sin β t C1′ e iβt + C2′ e − iβt = (C1′ + C2′ )cos β t + i(C1′ − C2′ )sin β t C1′ e iβt + C2′ e − iβt = C1 cos β t + iC2 sin β t y = e αt[C1 cos βt + iC2 sin βt] The expression for y is now a bit more understandable than the original expression, but it still contains a complex term.

Most of the important information about the system response can be obtained from these roots. Classical Solutions of Ordinary Linear Differential Equations 39 The relevant questions about the response are the following: • Is the response stable? That is, will the response remain bounded when forced by a bounded input? • Is the response monotonic or oscillatory? • If monotonic and stable, how long will it take for the transients to die out? • If oscillatory, what is the period of oscillation and how long will it take for the oscillations to die out?

One should use a particular solution that is of the same order as the forcing function polynomial and include all lower terms, whether they are in the forcing function or not. For instance, we would use yP = A2t2 + A1t + A0 for each of the following forcing functions 12t – 72t2, 12 – 72t2, 12 – 12t – 72t2, and –72t2. 25. 25. 1, yP = A0 e3t, does not work because it is part of the homogeneous solution (try using it and see why it does not work). 306. 143t e3t Before presenting more examples, it is worthwhile to point out that all three last examples have the same corresponding homogeneous equation, and therefore, the form of the complementary solution yH is exactly the same.

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