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Second and Higher Order Linear Differential Equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
The method of undetermined coefficients is a special method of practical interest that allows one to find a particular solution of a nonhomogeneous differential equation. It is based on guessing the form of the particular solution, but with coefficients left unspecified. This method can be applied if Equation (4.7.11) is a linear differential equation with constant coefficients;the nonhomogeneous term f(x) can be broken down into the sum of either a polynomial, an exponential, a sine or cosine, or some product of these functions. In other words, f(x) is a solution of a homogeneous differential equation with constant coefficients, ψ[D]f=0.
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
where yh(x) is the general solution of the associated homogeneous differential equation (4.7.2), and yp(x), which is a particular solution to a nonhomogeneous differential equation. The general solution yh(x) of the homogeneous equation (4.7.4) is frequently referred to as a complementary function. The method of undetermined coefficients is a special method of practical interest that allows one to find a particular solution of a nonhomogeneous differential equation. It is based on guessing the form of the particular solution, but with coefficients left unspecified. This method can be applied if
The z-transform, difference equations, and discrete systems
Published in Alexander D. Poularikas, ®, 2018
The particular solution of a nonhomogeneous equation is the method of undetermined coefficients. The method is particularly efficient for input functions that are linear combinations of the following functions: nk, where n is a positive integer or zeroan, where a is a nonzero constantcos an, where a is a nonzero constantsin an, where a is a nonzero constantA product (finite) of two or more functions of type 1 to 4
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.