An Analysis of Modified Emden-Type Equation ẍ + αxẋ + βx 3 = 0 : Exact Explicit Analytical Solution , Lagrangian , Hamiltonian for Arbitrary Values of α and β

The modified Emden-type is being investigated by mathematicians as well as physicists for about a century. However, there exist no exact explicit solution of this equation, x xx x   3 0 + + = α β for arbitrary values of α and β. In this work, the exact analytical explicit solution of modified Emden-type (MEE) equation is derived for arbitrary values of α and β. The Lagrangian and Hamiltonian of MEE are also worked out. The solution is also utilized to find exact explicit analytical solution of Force-free Duffing oscillator-type equation. And exact explicit analytical solution of two-dimensional Lotka-Volterra System is also worked out.


INTRODUCTION
The modified Emden-type equation [MEE] or the modified Painleve-Ince equation is where the dot represents derivative with respect to time and α, β are arbitrary parameters.The equation has been investigated by a lot of physicists as well as mathematicians [1][2][3].Painleve, himself obtained solutions of (1.1) for α 2 = 9β and α 2 = -β.This equation is extremely important because it arises in many mathematical problems like univalent functions defined by differential equation of second order [4].This equation also arises in modelling of fusion of pellets [5].The MEE is a special case of one dimensional analogue [6,7] of gauze boson theory introduced by Yang and Mills.In the study of equilibrium configuration of spherical gas clouds [8,9], this equation is also encountered.
As regards solution of MEE i.e., Equation (1.1), some progress in recent past has been done by Chandrasekar et al. [10].These authors have made good contributions by finding Lagrangian, Hamiltonian and Invariant of (1.1) for distinct cases: 1) α 2 = 8β, 2) α 2 > 8β and 3) α 2 < 8β.However the solution, they have found, is unsatisfactory.In this paper, exact analytical explicit single solution of (1.1) will be shown valid for arbitrary values of α and β.Also single Lagrangian as well as Hamilton valid for all α and β will be worked out.

LAGRANGIAN AND HAMILTONIAN OF MEE
In this section we will find the Lagrangian and Hamiltonian using a method due to Vujanovic and Jones [11].These authors have shown after a long calculation that the Lagrangian of the equation: The dot and prime represent derivative with respect to t and x respectively.
Is given by  i.e., ( ) Equation (3.9) is an algebraic quadric in n whose solution is: and also from (3.6) where A 0 and n are given by (3.11) and (3.10) respectively.This Lagrangian is valid for arbitrary values of α and β.

SOME SPECIAL CASES
Cariena and others [12] has considered MEE (1.1) with α = 3K and β = K 2 and has shown that for this choice the Lagrangian is For α = 3K and β = K 2 another Lagrangian of (1.1) is ( ) In this section we will show that both the above Lagrangians (4.1) and (4.2) follow from our derived general expression for Lagrangian of MEE (3.12) For, if we take α = 3K and β = K 2 , we find from (3.10) Then for n = −1 and A 0 = 2K we get from (3.12) And for n = 1/2 and A 0 = K one finds from (3.12) Rejecting the factor 2 in the denominator of R. H. S of (4.6) we get Thus it follows from above that our general expression for Lagrangtian of (1.1) correctly produce Lagrangian of (1.1) for α = 3K and β = K 2 , derived earlier by other authors.
The above analysis and examples clearly indicate that the Lagrangian of modified Emden-type Equation (1.1) can be derived for arbitrary values of α and β from Equation (3.12), which is the general uniform expression for Lagrangian of (1.1).

HAMILTONIAN OF MODIFIED EMDEN-TYPE EQUATION
The calculation of Hamiltonian of (1.1) is straight forward.From (3.12) and 2) and (3.12) i.e., Equation (5.4) is therefore, the expression for Hamiltonian of modified Emden-type Equation (1.1) for arbitrary α, β, where n and A 0 are given by Equations (3.10) and (3.11) respectively.

SOME APPLICATIONS
The specific cases of (6.1) for α = 0 and β = 0 have been studied by many authors [13][14][15].However their results are for specific cases only.
The solution of (6.1) for arbitrary α, β and γ can be obtained in a straight forward manner from our results of earlier sections in the following way: An invertible point transformation [10]  , , b b  etc are real parameters.This equation has been studied for a long time and its solution is important in mathematical biology [16].
For b 1 = a 1 and b 3 = -a 3 Equation (6.4) is of the form ( ) ( ) For these choices of α, β and γ solution of (6.6) follows directly from (6.3): Therefore from (6.8) to (6.12) Equation (6.7) and (6.13) give explicit analytical solution of Lotka-Volterra Equation (6.5).It is to be mentioned here that these solutions of Lotka-Volterra equation are perfectly new addition to literature.

CONCLUSION
In the above work, the initially proposed things, i.e., analytic explicit solution of Emden-type Equation (1.1) for arbitrary α, β and the Lagrangian and Hamiltonian of Emden-type equation are all worked out.Further, as applications of the results, the Force-free Duffing oscillator and Lotka-Volterra (two dimensional) equation is also analytically solved.All the solutions are explicit and can be directly applied.It turns out that all the results obtained in this paper are new addition to literature.

3 , 5 )
x x a a x a y y y a b x a y can be written by eliminating y and y  as