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: strategy for fixing First-Order traditional Differential Equations again to part 2-5Equations, Unknowns, and levels of FreedomClassical recommendations 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: procedure 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 suggestions of normal Linear Differential EquationsExamples of Differential EquationsDefinition of a Linear Differential EquationIntegrating issue MethodCharacteristic Equation
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Additional info for A First Course in Differential Equations, Modeling, and Simulation
8 h = ρf1 nonlinear because of the second dt term on the left-hand side of the equal sign. 3 Integrating Factor Method The integrating factor method provides a solution to any first-order linear differential equation. 10) dx where P(x) = a0(x)/a1(x) and Q(x) = r(x)/a1(x). 11) y=e ∫ Q( x) e The integration constant C is obtained using the initial condition. 4, dv = − g with v (0) = v i dt dv + (0)v = − g dt dv + P(t )v = Q(t ) dt where P(t) = 0 and Q(t) = –g. 14a dv = − g − rv dt with v (0) = v i dv + rv = − g dt dv + P(t )v = Q(t ) dt where P(t) = r and Q(t) = –g.
Note that the mass of the object is not specified this time. 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.
24b) 6 equations, 6 unknowns There is now a zero (0) DoF, and we can solve the model. In the next chapters we will encounter many other examples and realize how the procedure just explained helps in the modeling effort. 8 Summary This chapter is rather short, but hopefully it was easy to read and provided more motivation to read and study the book. The chapter clearly showed that the modeling starts from a basic law and that experimental facts are used to complement the development. The procedure of counting equations and unknowns, while developing a model, was presented and it was shown why it is helpful to do so.
A First Course in Differential Equations, Modeling, and Simulation by Carlos A. Smith