Pell equation solution set

Can we guarantee that Pell's equation has infinite solutions in positive integers without finding a non-trivial solution?

Linear equation system with Javascript

I am needing help with writing the code in a Javascript function to solve a system of equations of first degree, either any method (substitution, equalization or reduction), my idea is for a page to enter the numbers "n" x + "n" y = "n" (with 2 equations) and let the user touch the "Calculate" button and show the result for x and y.
Thanks in advance.

Finding the equation of the circle.

My question is how to find the eq. of the circle that passes through (0,1) and concentric with the circle centered in (2,1) and radius 3

Vortex motion equation for flow flow instability

I am solving the equation of motion of an isolated vortex oscillation in a superconductor. I am assuming that the driving force is generated by an RF current that oscillates at 1.3 GHz.
The main point of this calculation is to see what the effect of the instability of the flow of flow in the RF unit is.
The code that I wrote is reported below.

f = 1.3 10 ^ 9; (* Hz *)
[Omega] = 2. Pi f;
Trf = 1./f;

Tc = 9.25; (* K *)
T = 1.5; (* K *)

[Mu]0 = 4. Pi 10 ^ -7; (* H / m *)
[Phi]0 = 2.07 10 ^ -15; (* Wb *)
n = 5.56 10 ^ 28; (* m ^ -3 *)
e = 1.6 10 ^ -19; (*DO*)
m = 9.1 10 ^ -31; (* kg *)

[Tau] = 0.5 10 ^ -9; (* s *)
vf = 1.37 10 ^ 6; (*Mrs*)
Bc2 = 410. 10 ^ -3; (* T *)
l = 100 10 ^ -9; (*subway*)
[Lambda] = 49.5 10 ^ -9; (*subway*)
[Xi] = 28 10 ^ -9; (*subway*)
[Kappa] = [Lambda]/ [Xi];

[Gamma] = ([Phi]0) / ([Mu]0 [Lambda] );

v0 = Sqrt[(l vf Sqrt[14 Zeta[3] (1 - T / Tc)]) / (3 Pi [Tau])];

[Rho] = (m vf) / (n e ^ 2 l);
[Eta] = ([Phi]0 Bc2) / [Rho];

g = Log[[[[[Kappa]]+ 0.5 + Exp.[-04-08Log[[-04-08Log[[-04-08Log[[-04-08Log[[Kappa]]-0.1 (Log[[[[[Kappa]]) ^ 2];
[Epsilon] = ([Phi]0 ^ 2 g) / (4. Pi [Mu]0 [Lambda]^ 2);

Tmax = 3. Trf;
Zmax = 1. 10 ^ -6; (*subway*)

B = 100. 10 ^ -3; (* T *)

sol = NDSolve[{
(v0 ^ 2 [Eta]) / (v0 ^ 2 + D[x[t, z], t]^ 2) D[x[t, z], t]==[Epsilon] re[x[t, z], z, z]+ [Gamma] B Cos[[[[[Omega] t]Exp[-(z/[-(z/[-(z/[-(z/[Lambda])],
X[0., z] == 0.,
X[t, Zmax] == 0.,
(RE[x[t, z], z]/. z -> 0) == 0.
}, x, {t, 0., Tmax}, {z, 0., Zmax},
AccuracyGoal -> 13,
PrecisionGoal -> 2,
MaxSteps -> Infinity];
you[t_, z_] = Evaluate[x[t, z] /. Sun][[1]];
du[t_, z_] = D[u[t, z], t];

phaseSpace =
Parallel table[{Re[u[t, 0.]]10 ^ 6, Re[du[t, 0.]]10 ^ -3}, {t, Trf,
Tmax, Tmax / 500.}];
ListPlot[phaseSpace, Joined -> True, PlotRange -> All]

I can get an almost correct result from this calculation only if I fix the domain of the solution to 1 micron (Zmax = 1. 10 ^ -6). However, this is not correct because the vortex should be able to oscillate more than that within the volume, therefore, literally, I am cutting my solution using too small a domain.

If I fix Zmax to a larger number (for example, 5 um), I need to correct MaxStepSize to a small number to avoid errors and the calculation would take more than 2 days on my PC (actually I stopped it after 2 days because the PC froze !). Does anyone know how to solve this problem?

Also, I was wondering if there is any way to solve the equation when the velocity of the vortex is greater than v0 (the start of the velocity for the instability of the flow of flow). If I fix B to say 200 mT, then the velocity of the vortex would be greater than v0 and the calculation will not yield a correct solution, just a series of peaks. Is there a method I can use to solve this problem?

Please let me know!
Thanks in advance.

Mattia

Variation of the biharmonic equation with Neumann conditions.

I am currently writing a script to plot the solution of a variant of the biharmonic equation. In this case, the equation I want to solve is

Laplacian[[Alpha] Laplacian[u[x,y], {x, y}], {x, y}]+ [Beta]Laplacian[u[x,y], {x, y}]+ [Gamma] you[x,y] == 0

where [Alpha], [Beta]Y [Gamma] Constants are prescribed. I have been able to trace a solution successfully using the following:

biharm = {laplaciano[u[x, y], {x, y}]== v[x, y],
Laplacian[[Alpha] v[x, y], {x, y}]== - [Beta] v[x, y] - [Gamma] you[x, y]};
linked = {u[0, y] == 0, u[1, y] == 0, u[x, 0] == 0, u[x, 1] == 0,
v[0, y] == 0, v[1, y] == 0.5, v[x, 0] == 0, v[x, 1] == 0};
sol = NDSolveValue[{biharm, bound}, u, {x, 0, 1}, {y, 0, 1}];
Plot3D[Sun[Sun[Sol[sol[x, y], {x, 0, 1}, {y, 0, 1}, PlotRange -> All]

However, the boundary conditions with v[x,y] They are not really I want that I want to use; Instead, I would like to prescribe conditions for re[u[x,y], X] Y re[u[x,y], Y] in the limits of my region as I have done in the first line of boundary conditions.

When I try to implement this, I receive the following error:

The dependent variable in ... in the contour condition Dirichlet condition[...] It needs to be linear.

From there, the script takes a blank space. 3DPlot. The error seems to indicate that I would be able to specify something like NeumannValue instead of Condition of Dirichlet, but none of the similar examples I've seen seems to indicate how I can do that in my case. (For example, I've seen NeumannValue[...] done with the Laplace equation, but I'm not sure how to implement it here.)

Is there any way to do this here, or have I missed something in the other examples?

precalculus algebra: solve the equation on reals: $ sqrt {5x ^ 2 + 27x + 25} – 5 sqrt {x + 1} = sqrt {x ^ 2 – 4} $.

Solve the equation over reais: $ sqrt {5x ^ 2 + 27x + 25} – 5 sqrt {x + 1} = sqrt {x ^ 2 – 4} $.

This problem is an adaptation of a recent competition. And I can not solve it.

The solutions are $ sqrt {5} + 1 $ Y $ dfrac {13 + sqrt {65}} {8} $, as Wolfram Alpha says.

Problem in solving the differential equation.

Can someone tell me why I have errors when I try the following code?

`cd
s0 = 10; u = 0.4; w = 0.9; v = 0.01; k = 0.1; p = 5; L = 4; (* I choose them randomly *)
pde1 = w * D[c[x, t], t]+ u * D[c[x, t], X]- v * D[c[x, t], {x, 2}]== -k * c[x, t]* s[x, t];
pde2 = D[s[s[s[s[x, t], t]== -p * k * c[x, t]* s[x, t];
sol = NDSolve[{pde1, pde2, c[0, t] == cd

Scattered Arrangements: I want to solve this equation using a LinearSolve but I can not solve it

S = 10;
TH = 6;
[Mu]e = 60;
[Mu]g = 30;
[Lambda]e = 30;
[Lambda]g = 90;
CleanSlate[];
condition[S_, TH_][i_, j_]    : = p[i, j] Boole @ Or

The
j == 0 && i == S,
j == 0 && TH <i && i <S,
j == 0 && i == TH,
j == S - i && S - TH <= i && i <S,
j == TH - i && 0 <i && i <TH,
j == TH && i == 0,
j == TH && 0 <i && i <S - TH,
0 <j && j <TH && (TH - j) <i && i < (S - j)];
variable = SparseArray[{i_, j_} :> condition[S, TH][i - 1, j - 1], {S + 1, TH + 1}]// Flatten
u = Total[variable, 2];
seikikasiki = {u == 1};

soda[i_, j_] : = [Mu]e * p[i, j] ==[Lambda]e * p[i -1, j] /; j == 0 && i == S (* No1 *)
soda[i_, j_] : = ([Lambda]e + [Mu]my)*
P[i, j] ==[Mu]e * p[i + 1, j] + [Mu]g * p[i, j + 1] + [Lambda]e * p[i - 1, j] /; j == 0 && TH <i && i <S (* No2 *)
soda[i_, j_] : = ([Lambda]e + [Mu]my)*
P[i, j] ==[Mu]e * p[i + 1, j] + [Mu]g * p[i, j + 1] /; j == 0 && i == TH (* No3 *)
soda[i_, j_] : = ([Mu]e + [Mu]g) * p[i, j] ==[Lambda]e * p[i - 1, j] /; j == S - i && S - TH <= i && i <S (* No4 *)
soda[i_, j_] : = ([Lambda]e + [Mu]e + [Mu]Sun)*
P[i, j] ==[Mu]e * p[i + 1, j - 1] + [Mu]e * p[i + 1, j] + [Mu]g * p[i, j + 1] + [Lambda]Sun[i, j - 1] /; j == TH - i && 0 <i && i <TH (* No5 *)
soda[i_, j_] : = ([Lambda]e + [Mu]g) * p[i, j] ==[Mu]e * p[i + 1, j - 1] + [Mu]e * p[i + 1, j] + [Lambda]Sun[i, j - 1] /; j == TH && i == 0 (* No6 *)
soda[i_, j_] : = ([Lambda]e + [Mu]e + [Mu]Sun)*
P[i, j] ==[Mu]e * p[i + 1, j] + [Lambda]e * p[i - 1, j] /; j == TH && 0 <i && i <S - TH (* No7 *)
soda[i_, j_] : = ([Lambda]e + [Mu]e + [Mu]Sun)*
P[i, j] ==[Mu]e * p[i + 1, j] + [Lambda]e * p[i - 1, j] + [Mu]g * p[i, j + 1] /; 0 <j && j <TH && (TH - j) <i && i <(S - j) (* No8 *)
soda[i_, j_] : = 0
k = SparseArray[Table

soda[Table[soda[Table[sosa[i, j], {i, 0, S}, {j, 0, TH}]]// Flatten;
eqn = DeleteCases[k, 0] // Flatten;
sosasiki = Join[eqn, seikikasiki] // Flatten;
ansss = DeleteCases[sosasiki, 0] // Flatten
ans = LinearSolve[ansss, variable] // Flatten

I want to calculate with this program, but I can not calculate well. Are the elements of the matrix different, different expressions or the use of different LinearSolve?
From now on, I want to solve using S and TH as large values, and I want to calculate with LinearSolve considering the efficiency of the memory.
Please, give me some advice. Thank you.

Difficulties with NDSlve close to 0, Laplace-Helmholtz equation

I am using a tanh to solve in an interval Laplace equation for my variable and the Helmholtz equation in another interval.

Sense $ Delta n = -n $ in an interval and $ Delta n = 0 $ on the other, with an interface. In my case, the Laplacian is a spherical Laplacian with a radial symmetry.

Here is my code:

R = 1
F[x_, R_] = - (Tanh[(x - 1)/0.05] - R) / 2
eps = 10 ^ (- 5)
NDSolve[{(2 x Derivative[1][n][x]    + x ^ 2 (n ^ [Prime][Prime])[x]) /
x ^ 2 == 1/2 n[x] (-1 + Tanh[20.` (-1 + x)]), n[10*R] == 0,
north & # 39;[eps] == eps}, f, {x, eps, 5 * R}]Plot[{f[x, R], n[x] /. sun}, {x, 0, 3 R}]

But I get errors:

NDSolve :: dsvar: 0.00006128571428571428` can not be used as a variable.

That I have to do ?

2d – Why is the equation $ S = ut + frac 1 2 to t ^ 2 $ not used directly to calculate the final position of an object?

Most of the articles that I have read about the integration of physics in games say:

  1. First we calculate the speed after a certain elapsed time dt I like this:
    velocity.x + = acceleration.x * dt

  2. then we calculate the position as: x + = velocity.x * dt

But this will definitely give an inaccurate result.

Therefore, what is the reason for not using the equation of motion? $ S = ut + frac 1 2 to t ^ 2 $ To calculate a precise displacement and add it to the position?