# Solving a system of two equations of third degree

I am trying to solve the following system of two equations for x and y. The solutions for x and y have to be positive and the parameters s and c are positive (to have a coherent interpretation of my problem). Could anyone please help me?

eq1 := (2 + 12*s)/(9*x^2) + (4*Sqrt[2*s])/(9*x^2*y) - (4*Sqrt[2*s])/( 9*x^3) - k

eq2 := (1 + 3*Sqrt[2*s] - 6*s)/(9*y^2) + (2*Sqrt[2*s])/(9*y^2*x) - (2*Sqrt[2*s])/(9*y^3) - k

Solve[{eq1 == 0, eq2 == 0}, {x, y}, Positive]

-
Positive is not a legitimate domain as the 3rd argument to Solve. What, exactly, is intended to be positive -- just the desired values of x and y? If so, how do you expect Mathematica to do that when you don't say anything about the values of k? (Or by 'c do you actually mean k?) – murray Apr 8 '14 at 16:55
If you multiply each equation by the greatest common denominator of its terms, you'll see that in fact you have a pair of fourth degree equations, not third degree. – murray Apr 8 '14 at 17:00
Let's try an example: s = 4; k = 1;. Then Reduce[{eq1 == 0, eq2 == 0}, {x, y}, Reals] yields a set of solutions involving Root expressions for eight degree polynomials! – murray Apr 8 '14 at 17:07
I doubt you'll be able to enforce positivity of solutions without specifying actual values for the parameters. – Daniel Lichtblau Apr 8 '14 at 21:34

Here is a way of finding the general solution of your equations:

Define the expressions

expr1 = (2 + 12*s)/(9*x^2) + (4*Sqrt[2*s])/(9*x^2*y) - (4*Sqrt[2*s])/(9*x^3) - k;
expr2 = (1 + 3*Sqrt[2*s] - 6*s)/(9*y^2) + (2*Sqrt[2*s])/(9*y^2*x) - (2*Sqrt[2*s])/(9*y^3) - k;


Solve for y in terms of x.

ysol = Solve[expr1 == 0, y]

(* {{y -> (4 Sqrt[2] Sqrt[s] x)/(4 Sqrt[2] Sqrt[s] - 2 x - 12 s x + 9 k x^3)}} *)


Solve for x.

xsol = Solve[expr2 == 0 /. ysol, x]

(* <large Root expressions> *)


Verify the solutions.

{expr1, expr2} /. ysol /. xsol // FullSimplify

(* {{{0, 0}}, {{0, 0}}, {{0, 0}}, {{0, 0}}, {{0, 0}}, {{0, 0}}, {{0, 0}}, {{0, 0}}} *)

-
But this does still not give any closed-form explicit solution, as the solution still involves Root objects. – murray Apr 8 '14 at 20:52
It depends what you mean by "closed-form explicit"! I sense that you actually want a solution expressed in terms of "elementary" functions, but in this case you are out of luck because the polynomial degree is too high. – Stephen Luttrell Apr 8 '14 at 21:58
I suspect that the O.P. did expect to obtain such a "closed-form explicit". While I'm well aware of Abel's Theorem prohibiting a "general formula" for higher-degree polynomial equations, particular higher-order polynomial equations can be solved, and it might have been the case that the particular system about which the O.P. asked would lead to such. – murray Apr 9 '14 at 3:31
Many thanks for your insights. The variables (x and y) and parameters (s and c) have to be positive. I think will not be able to get a general solution for this system. – user13544 Apr 9 '14 at 11:35
Please don't add "thank you" as an answer. Once you have sufficient reputation, you will be able to vote up questions and answers that you found helpful. – Dr. belisarius Apr 9 '14 at 12:01