Ordinary Differential Equations and Applications I: With Maple Examples

This section introduces the sensing and perception of the soft robots. First, the sensor of the soft robots are classified. Then, the state of the soft robot’s sensing and perception model is introduced, i.e., passive mode, semi-active mode, active mode and interactive perceptive mode. Moreover, according to the sensing and perception mode, several sensing and perception technologies are presented. In the end, several challenges regarding the sensing and perception of the soft robots are summarized.

In this chapter, first order linear differential equations together with the nonlinear Bernoulli equations are considered and methods for obtaining their solutions discussed. The methods are applied to find solutions to models in the vibration of LR electrical circuits, radioactive dating, population dynamics, chemical reactions, epidemiological and pollution problems.

Second order differential equations and methods for solving them are studied. Methods considered are: Undetermined coefficients, Green’s and Wronskian, principle of superposition of solutions and variation of constant parameters. Also elucidated upon are: Construction of Green’s functions and applications to boundary value problems, Cauchy-Euler, Lagrange and Clairaut equations. Many solved examples and presents which include Maple ones.

In this chapter, Fourier series is introduced for functions which are Riemann integrable and are of bounded exponential growth. Orthogonal relations; least square error, completeness relation and Riemann- Lebesgue theorem are also considered. The Fourier series is applied to obtain a series solution to some periodic boundary value problems. Also provided are Maple examples for applications of Fourier series to ordinary differential equations.

Laplace transform method is studied in this chapter. It is an operation calculus method for finding solutions to differential equations, especially solutions to models in engineering. Problems solved using the Laplace transform are: LRC electrical problems with constant voltages and n-th order linear differential equations. In real life, systems may be governed by combinations of continuous and discrete characteristics, referred to as hybrid systems. In order to handle such systems effectively, the Laplace transform of discrete systems is also studied to complement the continuous systems.

 Linear space structures through vector space and inner products are examined. The D’ operator method, Cauchy-Schwartz inequality, is used. ‘The D’ operator is applied to obtain solutions to some practical problems.

In this Chapter, the power series method for generating linearly independent solutions to ordinary differential equations is considered. The method is applied to the Bessel, Hypergeometric, Legendre and Airy equations. Some special topics for transforming nonlinear equations to linear ones by the change of variables are considered, including corresponding Maple examples for obtaining symbolic and numeric solutions to ordinary differential equations using power series and other special functions.