# System of non-linear equations - Symbolic solution

I've been running this code for more than 12 hours and still can't find a symbolic solution for this. Maybe there is a way to simplify it or something might be wrong with the code. Do someone have a clue?

F1[un] = gamma0 + gamma1*un; F2[un] = sn*pin*un; F3[us] = delta0 + delta1*us + delta2*p; F4[us] = ss*pis*us; F5[us, un] = thetan*((p)^(mus - 1))*((uskss + ksn)^(es)) - thetas*(pi)^(-mun)*((un*kn)^(en)); eqns = {F1[un] == F2[un] - F5[us, un], F3[us] == F4[us] + F5[us, un] }; soln = Solve[eqns, {us, un}]


As it is clear from the code, I'm trying to get a symbolic solution (function of the parameters) for "Us" and "Un". I do need to use a symbolic solution, a numerical one cannot help me, unfortunately.

• Numericize your coefficients and use a nuemrical solver instead. You would not be able to read the output after 12 hour anyways (for its immense complexity). – Henrik Schumacher Apr 2 at 16:57
• Did you mean F1[un_] not F1[un], etc.? Also did you intend pi to be the constant Pi? – user55405 Apr 2 at 17:38

Edit By the way i corrected your typos. Function definition needs pattern objects like F1[un_]= ...

The critical parameter is en. Insert rational numbers for en to get solutions. The more simple en, the more simple the solution. I don't show results here. Try!

Solve[eqns /. en -> 1, {un, us}]

Solve[eqns /. en -> 2, {un, us}]

Solve[eqns /. en -> 3/2, {un, us}]

Solve[eqns /. en -> 6/7, {un, us}]


I guess something is wrong about the way you wrote "F5" equation. Shouldn't it be written as below?

F5[us, un] =
thetan*((p)^(m*us - 1))*((us*kss + ksn)^(es)) -
thetas*(pi)^(-mun)*((un*kn)^(en))


Because your "F5" it is actually not a function of "us"

Let's do some substitutions to simplify the forest of variables that you have

c1==gamma0+thetan*p^(mus-1)*(uskss+ksn)^es
c2==gamma1-sn*pin
c3==-thetas*pi^-mun
c4==delta0+delta2*p+gamma0
c5==delta1*-ss*pis


Notice that F5 is large and complicated with lots of exponents and appears as F5 in one equation and -F5 in the next equation. Add the first equation to the second to eliminate one of the F5 and make the problem far far simpler.

All this reduces your system to

eqns={c1+c2*un==c3*(un*kn)^en,c4+c5*us==-c2*un};


Notice your first equation tells you what un is. Notice your last equation says if you know un that you immediately know us. I don't think that anyone could say either of those things given the way your problem was originally posed.

All this reduces your problem to

Reduce[c1+c2*un==c3*(un*kn)^en,un]


Now would you rather solve this problem or your original problem? Neither problem is easy, but at least this one has a chance of you ever finding a solution.

Please check all this very carefully to make certain I have made no mistakes.

Has this problem, or a problem very similar to this problem, been asked several times in perhaps several places in the last day or two? Two equations with complicated exponentials, one of which can be eliminated by adding the two equations seems like I have seen almost exactly this elsewhere in the last day or two.