Abstract
We present a simple method of solving a system of two first order linear homogeneous differential equations with variable co-efficients and demonstrate a correspondence of the system of equations with non-linear Riccati equation. By using appropriate substitution, this non-linear equation is converted into second order linear homogeneous differential equation, the solution of which is used to find the solution of the system of equations. It is seen that different sets of the variable co-efficients of the system of equations correspond to second order linear homogeneous differential equations of different nature. We have illustrated the method with one example. We have shown that the method can be applied to obtain the approximate analytical solutions of the spin dependent coupled Altarelli–Parisi evolution equation in Quantum Chromo Dynamics under plausible conditions.
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Simmons G.F.: Differential Equations. Tata McGraw-Hill, New Delhi (1979)
Goldberg J., Potter M.C.: Differential Equations (A system approach). Prentice Hall, Upper Saddle River, p. 07458
Derrick W.R., Grossman S.I.: Elementary Differential Equations. 4nd edn. Addison-Wesley, Boston (1996)
Rainville E.D., Bedient P.E.: Elementary Differential Equations. 7nd edn. Macmillan Publishing Company, New York (1989)
O’Connor, J.J., Robertson, E.F.: Jacopo Francesco Riccati. http://www-history.mcs.st-andrews.ac.uk/Biographies/Riccati.html (1996)
Fraga, E.S.: The Schrodinger and Riccati Equations, vol. 70. Lecture Notes in Chemistry. Springer, Berlin (1999)
Schwabl F.: Quantum Mechanics. Springer, Berlin (1992)
Shankar R.: Principles of Quantum Mechanics. Plenum, New York (1980)
Khare A., Sukhatme U.: Supersymmetry in Quantum Mechanics. World Scientific, Singapore (2001)
Atre, R., Panigrahi, P.K., Agarwal, G.S.: Class of Solitary wave solutions of the one-dimensional Gross-Pitaevskii equation, Phys. Rev. E (3) 73(5), 056611, 5 pp (2006)
Yang Q., Zhang H.J.: Matter-Wave Soliton solutions in Bose-Einstein condensates with arbitrary time varying scattering length in a time-dependent harmonic trap. Chin. J. Phys. 46, 457–459 (2008)
Abbas S., Bahuguna D.: Existence of solutions to quasilinear functional differential equations. Electron. J. Differ. Equ. 2009(164), 1–8 (2009)
Davis H.: Introduction to Nonlinear Differential and Integral Equations. Courier Dover Publications, New York (1962)
Ince E.L.: Ordinary Differential Equations. Dover Publications, New York (1956)
Polyanin A.D., Zaitsev V.F.: Handbook of Exact Solutions for Ordinary Differential Equations. 2nd edn. Chapman and Hall/ CRC, Boca Raton (2003)
Reid W.T.: Riccati Differential Equations. Academic Press, New York (1972)
Zwillinger D.: Handbook of Differential Equations. Academic Press, Boston (1989)
Bastami A.Al, Belic M.R., Petrovic N. Z.: Special solutions of the Riccati equation with applications to the Gross-Pitaevskii nonlinear PDE. Electron. J. Differ. Equ. 2010(66), 1-10 (2010).
Altarelli G., Parisi G.: Asyptotic freedom in parton language. Nucl. Phys. B 126, 298 (1977)
Altarelli G.: Partons in quantum chromodynamics. Phys. Rep. 81C, 1 (1982)
Halzen F., Martin A.D.: Quarks and Leptons. Wiley, New York (1984)
Gehrmann T., Stirling W.J.: Analytic approach to the evolution of polarised parton distributions at small x. Phys. Lett. B 365, 347–358 (1996)
Collins P.D.B.: An introduction to Regge theory and high energy physics. Cambridge University Press, Cambridge (1977)
Donnachie A., Landshoff P.V.: Perturbative QCD and Regge theory: closing the circle. Phys. Lett. B 533, 277 (2002)
Donnachie A., Landshoff P.V.: The proton’s gluon distribution. Phys. Lett. B 550, 160 (2002)
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Choudhury, D.K., Choudhury, R. System of First Order Linear Homogeneous Differential Equations (FLHE) and Riccati Equation. Differ Equ Dyn Syst 20, 127–137 (2012). https://doi.org/10.1007/s12591-012-0109-7
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DOI: https://doi.org/10.1007/s12591-012-0109-7