COMPARIONS OF SOLUTION OF LINEAR ODEs: LAPLACE TRANSFORM AND COMPUTATIONAL METHOD (original) (raw)
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Application of Laplace Transform in Solving Linear Differential Equations with Constant Coefficients
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In recent years, the interest in using Laplace transforms as a useful method to solve certain types of differential equations and integral equations has grown significantly. In addition, the applications of Laplace transform are closely related to some important parts of pure mathematics. Laplace transform is one of the methods for solving differential equations. This method is especially useful for solving inhomogeneous differential equations with constant coefficients and it has advantages compared to other methods of solving differential equations. Linear differential equations with constant coefficients are among the equations that can be solved using the Laplace transform. Because the transformation Laplace is one of the transformations that easily converts exponential functions, trigonometric functions, and logarithmic functions into algebraic functions. Therefore, it is considered a better method for solving linear differential equations with constant coefficients.
Generally it has been noticed that differential equation is solved typically. The Laplace transformation makes it easy to solve. The Laplace transformation is applied in different areas of science, engineering and technology. The Laplace transformation is applicable in so many fields. Laplace transformation is used in solving the time domain function by converting it into frequency domain. Here we have applied Laplace transformation in linear ordinary differential equations with constant coefficient and several ordinary equations wherein the coefficients are variable. Laplace transformation makes it easier to solve the problems in engineering applications and makes differential equations simple to solve. This paper presents a new technological approach to solve Ordinary differential equation with variable coefficient.
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In this paper, a relatively recent method, namely the differential transform method, is applied to devise a simple scheme for the determination of Laplace transforms. The approach exhibits simplicity in that, unlike the usual method which necessitates integration, it only entails easy differentiation and a few elementary operations. Illustrative examples are provided to demonstrate the applicability and efficiency of the technique.
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In this paper, a new method for solving ordinary differential equations is given by using the generalized Laplace transform L_n. Firstly, the authors introduce a differential operator δ that is called the δ-derivative. A relation between the L_n-transform of the δ-derivative of a function and the L n-transform of the function itself are derived. Then, the convolution theorem is proven. Using obtained theorems, a few initial-value problems for ordinary differential equations are solved as illustrations.
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Laplace transform is a very powerful mathematical tool applied in various areas of engineering and science. With the increasing complexity of engineering problems, Laplace transforms help in solving complex problems with a very simple approach just like the applications of transfer functions to solve ordinary differential equations. In this paper we present a new technological approach to solve Ordinary differential equation. The application of MATLAB to compute and visualize the Laplace transforms is also discussed.