By Carlos A. Smith

ISBN-10: 1439850887

ISBN-13: 9781439850886

IntroductionAn Introductory ExampleModelingDifferential EquationsForcing FunctionsBook ObjectivesObjects in a Gravitational FieldAn instance Antidifferentiation: strategy for fixing First-Order usual 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 ofRead 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: method for fixing First-Order usual Differential Equations again to part 2-5Equations, Unknowns, and levels of FreedomClassical ideas of normal Linear Differential EquationsExamples of Differential EquationsDefinition of a Linear Differential EquationIntegrating issue MethodCharacteristic Equation

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

**Example text**

17, we now have one more unknown, m. 21, respectively): Fg = –mg Fd = –Pvy 3 equations, 5 unknowns 4 equations, 5 unknowns For the model of the position of the object, we use again the definition of velocity, dy = vy dt 5 equations, 6 unknowns [y] At this point, we still have one degree of freedom. After thinking about it, we realize that we have not missed writing/considering any other equation and have not made any mistake counting equations and unknowns. Thus, we cannot solve the set of equations.

93 × 10 –3 A. For vS = 20 + 20 u(t) V, solve the model to obtain expressions for i1(t) and i2(t). 3 Classical Solutions of Ordinary Linear Differential Equations This chapter presents the classical solutions of ordinary linear differential equations; chapter 4 presents the Laplace transform method. Chapter 2 presented the methods of antidifferentiation and separation of variables for first-order ordinary differential equations; we thought it was instructional at that time to show the reader the solution of the models that were being developed.

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. Any constant multiple of a solution of a homogeneous linear equation is again a solution of that equation.

### A First Course in Differential Equations, Modeling, and Simulation by Carlos A. Smith

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