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Introducing Bernoulli Integral For Solving Some Physical Problems

Some of both modeling and problems in physics have been commonly presented in a first-order nonlinear differential equations (DE) of constant coefficients. Because the DE are integrable, therefore one must have an integral formulation for solving the physical problems. This paper introduces Bernoulli integral to complete the Tables of Integral for all of the Mathematical Handbooks.

Basically, the Bernoulli integral is integral form of the homogeneous Bernoulli differential equation (BDE) of constant coefficients. Under proper transformation, the Bernoulli integral can be used to generate another integral formulation especially for integrals that can be transformed into arctangent DE. By using the Bernoulli integral, one can create its self the integral formulation of solving the physical problems, and hence reduces utilization the tables of integral. A special application in generating Euler formula also presented.

Introduction

Some of both modeling and problems in physics have been commonly presented in a first-order nonlinear differential equations (DE) of constant coefficients For instance, in designing electromagnetic apparatus [Markus,1979], the logistic growth process [Welner,2004], chaotic behavior [Barger et al,1995], the generation and propagation of soliton [Wu et al 2005],[Morales,2005], the transport of fluxon [Gonzile et al,2006], the generation of squeezed laser [Friberg,1996 ],etc. One requires Table of Integral to solve a specific integral for solving such differential equation [Spiegel,MR,1968]. To complete the Table of integral, we introduce Bernoulli integral that until now not including in both of the Table integral and mathematical Handbook. By using the Bernoulli integral, one can create the integral formulation required in solving the physical problems, and hence reduces utilization the Tables of integral.

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4 Responses

1. on October 31, 2008 at 10:10 am | Reply Yono H.Pramono

Pak Rohedi…

Rumus ABC yang anda maksud memang menarik,
memang gampang sih merunut terjadinya rumus tersebut, namun ternyata dibalik solusi penemuan anda dengan metode Rohedi, lebih simple dan berlaku umum…
Selaksa memancar sinar untuk solusi polynomial orde yang lebih tinggi, misal solusi untuk pangkat tertinggi 3

Tangensial Hiperbolikus yang anda sajikan menggugah
seniman matematika yang sedang tidur….

Tokorode….mo nan nin gurai okyaku sama ga kimi no websaito wo miteta no?
(terjemah: ngomong2 sudah ada berapa pengunjung website rohedi.com sampai detik ini?)

Ja ne..

Yono H.Pramono
(Osaka Prefecture University, Sakai, Japan)

2. on December 18, 2008 at 5:50 pm | Reply Denaya

Is this Bernoulli Equation ?

$$\frac{dy}{dx}+px^3 y=qx^3y^3 +3$$

3. on December 22, 2008 at 1:02 pm | Reply afasmt

@Denaya,

The general Bernoulli differential equation (BDE) is of form
dy/dx + p(x)y = q(x)y^n, for n is not equal 1.

Hence, your nonlinear first order ODE

dy/dx + px^3y = qx^3y^3 + 3

is not BDE type, because there is number 3 as additional term at the right side. In mathematics literature your ODE is Chini DE’s type, but I call the ODE as inhomogeneous BDE. Of course you must have great effort to solve it.

4. on May 3, 2009 at 10:31 pm | Reply Nadya Fermega