By Carlos A. Smith
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: process for fixing First-Order usual Differential Equations again to part 2-5Equations, Unknowns, and levels of FreedomClassical suggestions of standard Linear Differential EquationsExamples of Differential EquationsDefinition of a Linear Differential EquationIntegrating issue MethodCharacteristic Equation. Read more...
summary: IntroductionAn Introductory ExampleModelingDifferential EquationsForcing FunctionsBook ObjectivesObjects in a Gravitational FieldAn instance Antidifferentiation: process for fixing First-Order usual 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 recommendations of normal Linear Differential EquationsExamples of Differential EquationsDefinition of a Linear Differential EquationIntegrating issue MethodCharacteristic Equation
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Extra info for A First Course in Differential Equations, Modeling, and Simulation
If oscillatory, what is the period of oscillation and how long will it take for the oscillations to die out? The important term of a stable response was just introduced in the above questions. We stress the definition just used, a response is stable if it remains bounded when forced by a bounded input. An unstable response is one that when forced by a bounded response, it continues moving up or down without stopping and reaching a final value; a stable response reaches a final value. Please note that the bounded input must be one that reaches a final value.
We can also express these last statements as Root α ± iβ Response is stable or unstable solely depending on the sign of α. If negative, the response is stable; if positive, the response is unstable Response is monotonic or oscillatory solely depending on the numerical value of β. 5 Roots of characteristic equation. The fact that a system response may be oscillatory or not does not have anything to do with its stability. The system may be stable or unstable (only depending on the sign of α) and may be oscillatory or monotonic (only depending on the numerical value of β).
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. Specifically, this chapter presents the definition of a linear differential equation followed by the methods of the integrating factor, characteristic equation, and undetermined coefficients.