By Carlos A. Smith

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: method 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: procedure 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 FreedomClassical options of standard Linear Differential EquationsExamples of Differential EquationsDefinition of a Linear Differential EquationIntegrating issue MethodCharacteristic Equation

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**Example text**

As mentioned above, and as we Classical Solutions of Ordinary Linear Differential Equations 45 shall see next, the form of the particular solution depends only on the form of the forcing function. It is somewhat more obvious now than previously to note that the solution of homogeneous differential equations is independent of the type of forcing function affecting the system. Thus, it is only dependent on the system itself, and it is why sometimes it is called the natural response. 24 as y = yH + yP Total solution depends only on the form of the forcing function (forced response) depends only on the system itself (natural response) We previously showed how to obtain the homogeneous solution using the characteristic equation method.

We will continue using this procedure throughout the book and strongly encourage the reader to do the same. 22) Thus, to solve any set of equations, DoF = 0! , DoF ≠ 0), then the engineer/designer has the “freedom” to select/specify the value(s) of any of the unknown(s) necessary to make DoF = 0. Obviously, the selection of what unknown(s) and its (their) value(s) must be done with common sense and engineering judgment. Let’s use the same example once more to explain/discuss this term of DoF in more detail.

In the case of objects in a gravitational field, the physical law that helps is Newton’s second law, F = ma where the arrows above the force F and the acceleration a indicate that these quantities are vectors (meaning that direction and magnitude must be specified). 1) The summation sign is used in Fy when there is more than one force acting on the object. In modeling the systems in this chapter, we have to choose a direction, up or down, as positive. 1 shows the set up. Develop the models that describe the velocity and position of the object as a function of time.