# Second and Higher Order Ordinary Differential Equations

01 Jan 2019-pp 91-153

TL;DR: Second-order ODEs explicitly contain a second derivative term, but no higher derivatives as mentioned in this paper, and the quantities of the second derivative may not appear explicitly in a second order ODE.

Abstract: Second-order ODEs explicitly contain a second derivative term, but no higher derivatives These equations are of the form \(F\left( {x,\,y,\,y^{\prime } ,y^{\prime\prime } } \right) = 0\) The quantities \(x,\,y,\,y^{\prime }\) may not appear explicitly in a second-order ODE, such as in the equation, \(y^{\prime \prime } = 3\)

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TL;DR: In this paper, a theoretical valuation formula for options is derived, based on the assumption that options are correctly priced in the market and it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks.

Abstract: If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks. Using this principle, a theoretical valuation formula for options is derived. Since almost all corporate liabilities can be viewed as combinations of options, the formula and the analysis that led to it are also applicable to corporate liabilities such as common stock, corporate bonds, and warrants. In particular, the formula can be used to derive the discount that should be applied to a corporate bond because of the possibility of default.

28,434 citations

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Academia Sinica

^{1}TL;DR: A very general framework for Cauchy-Euler equations is studied and an asymptotic theory of coefficients of functions satisfying such equations is proposed that covers almost all applications where CauCHY- Euler equations appear.

61 citations

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TL;DR: In this paper, a transform table similar to that used in Laplace transform theory is developed and applied to network problems, which can be described as: performing the integral operation, applying the table of operations on known transform pairs, and deriving the Hankel transform from the Laplace Transform.

Abstract: Integral transform techniques for solving linear integro-differential equations can provide insight and flexibility in solving physical problems, especially network problems. The type of differential equation which describes the physical system will dictate the transform that should be applied to simplify the solution and this paper deals with two transforms, namely, the Mellin transform and the Hankel transform. The Laplace transform can be used to solve linear constant coefficient differential equations or networks which are represented by this type of equation. A familiarity with this transform is assumed and is not covered in this paper. Mellin transforms may be applied to networks which yield the Euler-Cauchy differential equation. This transform will simplify the solution of such an equation. A transform table, similar to that type used in Laplace transform theory, is developed and applied to network problems. Hankel transforms may be applied to networks which yield the Bessel differential equation or variations of this equation. Unlike the Laplace and Mellin transforms, the Hankel transform is symmetric and the transformed variable is a real, rather than a complex variable. A transform table of both operations and functions is developed anti applied to network problems as before. Three methods can be used to establish the table of transform pairs. They can be described as: performing the integral operation, applying the table of operations on known transform pairs, and deriving the Hankel transform from the Laplace transform. With both transforms, the applications are made to problems in analysis, instrumentation, and synthesis.

37 citations

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01 Apr 1995

TL;DR: One-dimensional problems, separation of variables, and separation of Variables: as discussed by the authors The Laplace Transform Method and Perturbation Methods are two of the most commonly used methods.

Abstract: One-Dimensional Problems--Separation of Variables. Laplace Transform Method. Two and Three Dimensions. Green's Functions. Spherical Geometry. Fourier Transform Methods. Perturbation Methods. Generalizations and First Order Equations. Selected Topics. Appendices. References. Index.

20 citations