Alexey Ivanov

## 1130 Reputation

11 years, 211 days

## For example, Draghilev's method...

Draghilev's method  is a numerical-analytical method for solving underdetermined systems of equations.
In this case, we have an analytical solution.
PLAN_CURVE_3d_1.mw

## If I understood correctly...

```restart;
foo := proc (a, b) local x, y;
x := a^2-b; y := b^2-a;
x, y, " tests on return" end proc:
C := foo(6, 3):
a := `A,B&Assign;`:
print(a, C[1], C[2], C[3])```

## There is also a function specifically fo...

For example, such an option

```restart;
N := 60; L := 10; a := 2*10^(-12);
ns := [.9603, .9647, .9691];
eqn := proc (A) options operator, arrow; ns[i]-1+(N/A+(A*L)^(L/(1-L)))^((1-L)/L)*(2+a*exp((N/A+(A*L)^(L/(1-L)))^(1/L))/(A*L*(N/A+(A*L)^(L/(1-L)))^((L-1)/L)+a*exp((N/A+(A*L)^(L/(1-L)))^(1/L))))/(A*L) end proc:
for i to nops(ns)
do
RootFinding:-NextZero(eqn, 1.*10^(-10))
end do```

9.300682772*10^(-10)
3.054144321*10^(-9)
1.172662805*10^(-8)

## Where is the value of U0?...

```restart;
U0 := 110; #for instance
U := U0*exp(-(1/10)*t)*cos(2*t);
plot(U, t = -25 .. 50);
smartplot(U)```
```restart; U0 := 110;
U := t-> U0*exp(-(1/10)*t)*cos(2*t);
plot(U(t), t = -25 .. 50);
smartplot(U(t))```

## Maybe RootFinding [Isolate]?...

I suspect that we are talking about real solutions.

## It seems that it will be closer to the t...

```restart;
f := exp(t)+exp(-t^2-2*t);
g := exp(-t^2)+exp(-t^2-2*t);
solve([f-g], t);
plot([f, g], t = -3 .. 3, discont = true, color = [red, blue])```

You have one equation with three variables, It has a periodic solution in the form of inclined "tubes". Solution for a single "tube" can be obtained numerically using the Draghilev method, or another numerical method. After that, you can select the desired range for each variable from the resulting set of solutions.
In Maple Pi, not pi.

```restart; with(plots):
f := cos(2*Pi*(x+y-2*z))+cos(2*Pi*(y+z-2*x))+cos(2*Pi*(z+x-2*y));
#solve({f, 0 <= x, x <= 1, y <= 1, z <= 1, x < y, y < z}, [x, y, z], explicit);
implicitplot3d(f, x = -.5 .. .5, y = -.5 .. .5, z = -.5 .. .5, axes = normal, numpoints = 5000, color = blue, transparency = .5, style = surface, scaling = constrained, axes = boxed)```

## For example...

I remember that similar questions have already arisen many times.
For example, there is such a technique

`a := `(3*Pi)/10`; print(a)`

## For example, using Draghilev's method...

In this case, this method gives a parametric solution. In the text of the program, the Draghilev method is highlighted in green.
CURVE_3d.mw
(If satisfied with numerical parameterization, the examples can be very complex. But as far as analytical parameterization is concerned, even in this case, the method can perform an example more complicated.)

## And if you do this?...

solve(r2,{ksi});  or  fsolve(r2);

## transparency...

And if just combine "transparency" and "thickness"?
For example

```restart: with(plots):
f1:=x->0.95*x;
f2:=x->0.99*x;
f3:=x->0.991*x;
p1:=plot(f1(x),x=0..2,color=red,legend = f1(x)):
p2:=plot(f2(x),x=0..2,color=blue,legend = f2(x)):
p3:=plot(f3(x),x=0..2,color=green,thickness = 10, transparency = .7, legend = f3(x)):
display(p1,p2,p3);
```

## Draghilev method...

Just in case, search for "Draghilev method" (or Dragilev here:
https://www.maplesoft.com/applications/view.aspx?SID=149514 )
The method, in particular, finds continuous solutions of systems with free variables (underdetermined systems of equations).

## For instance...

You can do this:

``` restart:
a := plot({seq((6*x-2*t)/x^2, t = 1 .. 3)}, x = -1 .. 5, y = -1 .. 6):
b := plot(3/x, x = 0 .. 5, y = -1 .. 6, color = black, thickness = 3):
plots[display](a, b)```

## Because there are strict zeros...

Since childhood, I try to avoid strict zeros in coordinates (and generally avoid strictly identical values) due to formulas, because expressions may be nullified after substitution. It is better to shift the point o or d along the oX axis, for example:

```restart:
with(geometry):
point(o, 0.1e-11, 0.);
point(A, 0., 1.);
point(d, 0., 2.);
point(F, .8944271920, 1.4472135960);
line(lOD, [o, d]);
line(lAF, [A, F]);
alpha := FindAngle(lOD, lAF);
```

alpha = 1.107148718

## For example...

CompleteSquare(x^2+y^2-2*x-y-2 = 10, x);

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