one man

Alexey Ivanov

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12 years, 77 days

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It was found on the social networks of the WM group. Written in Python. Perhaps someone would like to adopt it.
 

The question is not at all from me, but, probably, one might say, from the authors of this publication.  interesting_system.pdf

Just for fun.
Find all real solutions to this 2x2 system of nonlinear equations in any given domain. 

f1:=x1-x1*sin(x1+5x2)-x2*cos(5x1-x2);
f2:=x2-x2*sin(5x1-3x2)+x1*cos(3x1+5x2);


 


 

 

Inscribed square problem


I decided to check on this curve 
 

 4*(x1-0.25)^4-x1^2*x2^2+(x2-0.25)^4-1.21=0;


I get a very rough solution, because the difference between the sides of the "square" begins already at 1-2 decimal places. More precisely, it doesn’t work, that is, we can say that I personally could not find confirmation of the hypothesis.
The coordinates of the vertices of the square:

                    -0.4823584672, -0.2770841741

                    0.9883885535, -0.3959790155

                    1.108267478, 1.086941264

                   -0.3459185869, 1.219514527

Side lengths:
                          1.475544911

                          1.487757882

                          1.460216690

                          1.502805215


 


Perhaps someone would like to try.

Another training example (number 2 and last) for finding all solutions to a system of equations:

f1 := x3^2-0.1*x1^4-0.05*x2^4+1;
f2 := x1^3+x2^3+0.05*x3^3-1; 
f3 := -2*cos(3*x1)+2*cos(3*x2)-2*cos(3*x3)+1;

In my version, there are 116 solutions.
Is it so?

This is a training example for finding all solutions to a system of equations. If you look at the graph, you can count 36 solutions, but I managed to find 20 relatively good approximations. No attempts to get more solutions, which are also visible as intersections of graphs, did not lead to success. Therefore, there is a suspicion that there are only 20 solutions.
Is it so?

 restart: with(plots):
 a:=8.:
 f1 := x1^4-1.999*x1^2*x2^2+x2^4-1; 
 f2 := tan(x1+x2)-x2*sin(x1);
 implicitplot([f1, f2], x1 = -a .. a, x2 = -a .. a, numpoints = 25000, scaling = constrained,  color = [red, blue], thickness = 1);
   1, (3.192246883291975), (-3.0395187374365404)
   2, (3.0952031367176476), (-3.2447717313041897)
   3, (0.5881900748267959), (-1.160066226905079)
   4, (-0.936866718243322), (-1.3700058362814254)
   5, (-2.555853694651265), (-2.7399958564861953)
  6, (-3.2556241416421168), (-3.3964651254113774)
   7, (-3.583319843955091), (-3.7077839724189228)
   8, (-5.364827188794712), (-5.401998918608201)
   9, (-5.398295356665546), (-5.360818510223991)
  10, (-3.769206506106412), (-3.6477855329362683)
  11,(-1.3978806247566642), (-0.9772190664843745)
   12, (-1.192159295544978),(0.6492335177657542)
   13, (-3.0867255059416623),(2.927375855548188)
    14, (-3.18519036357835), (3.329801919022179)
   15, (2.0108268901120754), (2.243492422396739)
   16, (3.1133812329649766), (3.261937603184373)
   17, (4.0265558604742715), (4.130826167761226)
    18, (4.124539552922121), (4.019977762680433)
    19, (3.172338340844501), (3.018365965761908)
   20, (2.1945695320368097), (1.9558412553082192)


 

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