solve nonlinear system of equation numerically

Question

solve the following system of equations numerically $$2x+2y - e^{xy} = 0$$ $$x^3 + y - xy^3 = 1$$

I'm also asked to solve analytically but I'm pretty sure the closed form solution doesn't exist because if yes, then the second equation would involve Cardano formula, plus the log/exponential form from the top, it doesn't look closed to me.

I know if there is a solution, then it shouldn't lie in the quadrant $x<0,y<0$ because from the first equation $x+y>0$ implies either $x>0$ or $y>0$. However, I don't know how to solve completely.

thanks for any help!

After consulting WolframAlpha it seems that there is one solution pair in each of the remaining quadrants... — Tintarn, Aug 25 '15 at 17:47
actually I would like to know how to solve the problem than finding the solution for it. I know MATLAB can fsolve() with no problem... — kensaii, Aug 27 '15 at 02:45

score 0 · Accepted Answer · answered Aug 25 '15 at 17:52

0

a numerical method is useful to solve this system, the only way that i have in mind is to solve the second equation for $x$ or $y$ and with the first equation you got an equation in $x$ or $y$ which you can solve also by the Newton method, i have found $$x \approx 1.059396315, y \approx -.8579048919$$

answered Aug 25 '15 at 17:52

Dr. Sonnhard Graubner

95,283

can you please explain to me how you set up an iterative Newton method? I don't see any way to substitute completely to reduce the number of variables. are you saying Newton method with 2 variables and Jacobian instead of simply first derivative? – kensaii Aug 25 '15 at 18:08
see here a script with example http://www.unc.edu/math/Faculty/pbe/publications/Newton.pdf – Dr. Sonnhard Graubner Aug 25 '15 at 18:10
oh thanks! that might be what I'm looking for. let me check! – kensaii Aug 25 '15 at 18:12
i have send you a link – Dr. Sonnhard Graubner Aug 25 '15 at 18:13
great! thank you very much for the link! – kensaii Aug 25 '15 at 18:18
i have found three solutions of the given system – Dr. Sonnhard Graubner Aug 25 '15 at 18:18

solve nonlinear system of equation numerically

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