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Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
A solution to a differential equation is called a particular (or specific) solution if it does not contain any arbitrary constant. By setting C to a certain value, we obtain a particular solution of the differential equation. So every specific value of C in the general solution identifies a particular solution or curve. Another way to specify a particular solution of y′=f(x,y) is to impose an initial condition: y(x0)=y0,
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
A solution to a differential equation is called a particular (or specific) solution if it does not contain any arbitrary constant. By setting C to a certain value, we obtain a particular solution of the differential equation. So every specific value of C in the general solution identifies a particular solution or curve. Another way to specify a particular solution of y'=f(x,y) is to impose an initial condition: yx0=y0,
Function Spaces
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
Separation of variables is a technique that may be used to find solutions to some differential equations. The separation we will be mainly concerned with here is the separation of partial differential equations, which essentially amounts to rewriting a differential equation in several variables as an ordinary differential equation in one of the variables and a partial differential equation in the remaining variables. Naturally, if there are only two variables in the problem, the separation will result in two ordinary differential equations. The idea is to reduce the problem to a set of simpler problems that may be solved individually. If we have a partial differential equation in the variables x, y, and z, we can attempt to separate the z coordinate by looking for solutions of the form () f(x,y,z)=g(x,y)Z(z).
Teaching transfer functions without the Laplace transform
Published in European Journal of Engineering Education, 2022
Imad Abou-Hayt, Bettina Dahl, Camilla Østerberg Rump
Regarding item (1), the curriculum of the course includes transfer functions of standard engineering systems, given that the concept ‘transfer function’ relies on the terminology of the Laplace transform. Looking at some standard textbooks on modelling and control dynamic systems, such as Ogata (1998) and Dorf and Bishop (2011), they start with a somewhat comprehensive introduction to the Laplace transforms before defining transfer functions. Accordingly, the students have to learn another method of solving differential equations, in addition to the one they have had in their mathematics course, namely, the method of undetermined coefficients (Boyce, DiPrima, and Meade 2017, p. 131). Moreover, this introduction to the Laplace transform involves decomposition into partial fractions in order to eventually arrive at the time-domain solution of the differential equation. This is another didactic variable to consider, given that partial fraction decomposition is an unduly tedious process.
Parameter identification for the simulation of the periodontal ligament during the initial phase of orthodontic tooth movement
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2021
Albert Heinrich Kaiser, Ludger Keilig, Reinhard Klein, Christoph Bourauel
A well known method for solving partial differential equations is separation of variables. Analogue, the model function for actuator force is assumed to be the product of a function of actuator displacement and a function of time with time and parameters of F and G: