       Finding a function that makes it possible to Solve[ ] a system of equations

• To: mathgroup at smc.vnet.net
• Subject: [mg114029] Finding a function that makes it possible to Solve[ ] a system of equations
• From: Mauricio Esteban Cuak <cuak2000 at gmail.com>
• Date: Sat, 20 Nov 2010 18:27:53 -0500 (EST)

```Hey MathGroup,

I apologise beforehand since this might turn to be more a mathematical query
than a Mathematica one.
However, I appreciate any suggestion on the matter.

In:= \$Version
Out= "7.0 for Mac OS X x86 (64-bit) (February 19, 2009)"

This is my setup:

f = (g*x + y)^h + d* i[x, y];
u1 = a*f - x^2;
u2 = (1 - a)*f - y^2;

Where x, y are the variables and {a,h,g,d} are just parameters, with 0 < a <
1, g > 0, d   bigger or equal to 0

i[x,y] is the function that I need to specify.

Further, it must be true that

i[ x, 0] = 0     &    i[ 0, y ] = 0

For simplicity I've set

h=1;

Though if the solution you suggest requires it, h could belong to (0,1]

What I need is the analytic solution for {x, y}, hopefully a unique real
solution (I don't mind if there are complex ones) to:

sol = Solve[{D[u1, x] == 0, D[u2, y] == 0}, {x, y}];

Obviously, I need to specify i[x,y] to do that. I need the function i[x,y]
to grow slower that x^2 or y^2.

i[ x, y] = x^(1/2)*y^(1/2)    or

i[x,y] = Log[1+x] * Log[1+y ]

do the trick but Mathematica can't find an analytic solution.

The only thing that works is i[x,y] = x*y but the problem is that it grows
as fast

as x^2.

Thanks for your time,

ME

```

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