2010 Issue №4

An application of the Abel equation of first type to the task of solving the Friedman equations

Pages

43-46

Abstract

An interesting connection between Einstein-Friedmann equations for the models of universe filled with scalar field and the special form of Abel equation of the first kind is presented. In particular, it is shown how, knowing the general solution of the Abel equation (corresponding to the given scalar field potential) one can obtain the general solution of the Friedmann Equation.

Reference

1. Abel N. H. Oeuvres Complétes II. S. Lie and L. Sylow, Eds., Christiana, 1881.

2. Appell P. Sur les invariants de quelques équations différentielles. // Journal de Math. (4). 5. (1889) 361—423.

3. R. Liouville. Sur une équation différentielle du premier ordre // Acta Math. 27 (1903) 55—78.

4. Murphy G. M. Ordinary Differential Equations and Their Solution // Prince­ton, NJ: Van Nostrand, 1960.

5. Zwillinger D. Handbook of Differential Equations, 3rd ed. // Boston, MA: Academic Press. 1997. P. 120.

6. Kamke E. Differentialgleichungen: Lösungsmethoden und Lösungen // Chel­sea Publishing Co, New York. 1959.

7. Sachdev P. L. A Compendium of Nonlinear Ordinary Differential Equations // John Wiley & Sons, 1997.

8. Chervon S. V., Zhuravlev V. M., Shchigolev V. K. New exact solutions in standard inflationary models // Phys. Lett. B398. (1997) 269—273.

9. Felder G., Frolov A. et al. Cosmology With Negative Potentials // Phys. Rev. D66 (2002) 023507.