Alexey Ivanov

## 1160 Reputation

12 years, 48 days

## Rose and Python...

Maple

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

## System of nonlinear equations 2x2...

Maple

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...

Maple

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.

## The second example of finding all soluti...

Maple

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?

## Example for finding all solutions to a s...

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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