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Definitions and Concepts
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
General solution Given an nth order ordinary differential equation, the general solution is a family of solutions depending on n arbitrary constants. For a linear differential equation, its general solution contains a linear combination of n linearly independent solutions. For example, the second order differential equation y′′+y=1 has the general solution y(x)=1+Asinx+Bcosx, where A and B are arbitrary constants.
Laplace Transform
Published in Nassir H. Sabah, Electric Circuits and Signals, 2017
The characteristic equation of a linear differential equation is a polynomial in s obtained by taking the LT of the equation with zero forcing function, zero initial conditions, and assuming a nonzero value of the variable of the equation. The LHS of the characteristic equation appears in all the responses derived from the differential equation.
Nonlinear free and forced vibration analysis of the sandwich composite cylindrical panel with auxetic core and GPL-reinforced facing sheets subjected to the temperature gradient
Published in Waves in Random and Complex Media, 2023
For Equation (31) is changed to the following linear differential equation: Equation (31) can be solved by using the following Tylor series expansion for Substituting Equation (35) into Equation (32) results in the following linear () and nonlinear () equation, respectively: The solution to Equation (36) is And is the nonlinear natural frequency of the sandwich panel. Substitution of Equation (38) into Equation (37) leads to the following equation: By setting secular terms equal to zero: From Equation (40) we have determined the nonlinear frequency of the sandwich panel: By setting in Equation (41):
Constant and quadratic damping of free oscillations: easy solutions
Published in International Journal of Mathematical Education in Science and Technology, 2022
Anastasios Adamopoulos, Nikolaos Adamopoulos
Equation (14) is a non-linear differential equation since the derivative of the unknown function appears at the power of two. There is no exact analytical solution to this equation. We will show that a numerical solution can be found using an Excel spreadsheet. From the definition of derivative we get: where is a sufficiently small increment of time. So, if we know , we can find using the above formula. But, we must also know . A similar expression for is the following:
1 V, 20 nW True RMS to DC Converter based on Third Order Dynamic Translinear Loop
Published in IETE Journal of Research, 2022
C. B. Muhammed Mansoor, Anuradha Patil, S. Rekha
Equation (6) is a non-linear differential equation. Thus, RMS–DC converter demonstrates the application of the dynamic translinear principle to the implementation of non-linear differential equations. In order to implement Equation (6) in the translinear domain, the derivative term in the equation is eliminated by introducing a capacitor current , as shown in Figure 2.