## Plotting – Plot the solution from DSolve

I try to solve a differential equation as in the following code:

``````FullSimplify(DSolve(x'(t) == a + b E^(g t) + (c + d E^(-g t)) x(t), x(t), t))
``````

what generates

Now I want to draw it with certain assigned parameter values, for example: `a = 1; b = 2; c = 3; d = 4; g = 0.1; A = 1` where I replaced the integration constant `c_1` With `A`,

Here is my code for plotting `x(t)`:

``````a = 1; b = 2; c = 3; d = 4; g = 0.1; A = 1
x(t_) := E^(-((d E^(-g t))/g) + c t) (A + Integrate(E^((d E^(-g K(1)))/g - c K(1)) (a + b E^(g K(1))), {K(1), 1, t}))
Plot(x(t), {t, 1, 10})
``````

It works forever. To check if Mathematica performs calculations, I tried

``````x(1)
``````

and it gave way

``````3.83926*10^-15
``````

which is nice. But when I tried

``````x(2)
``````

I have

It seems that Mathematica can only compute the integral if the integration region is $$int_1 ^ 1$$, Is that because the integrand is too complicated? Is there a way to let Mathematica calculate this? Many Thanks!

## Differential Equations – Using NDSolve on Wave PDE on String if the solution is given at 2 different times instead of the initial velocity?

This is a PDE from a Maple document. Mathematica DSolve can not solve it right now.

I wanted to review the Maple solution with NDSolve. This is a string of length 1, which is fixed on the left and can move freely to the right. Give a starting position and let go.

Here are the specifications of the PDE

Solve for $$0 0$$ the wave PDE
$$-u_ {tt} + u (x, t) = u_ {xx} + 2e ^ {- t} left (x – frac {1} {2} x ^ 2 + frac {1} {2} t – 1 right)$$

With boundary condition

begin {align *} u (0, t) & = 0 \ frac { partial u (1, t)} { partial x} & = 0 end {align *}

And initial conditions

begin {align *} u (x, 0) & = x ^ 2-2 x \ u (x, 1) & = u left (x, frac {1} {2} right) + e ^ {- 1} left ( frac {1} {2} x ^ 2-x right ) – left ( frac {3} {4} x ^ 2- frac {3} {2} x right) e ^ { frac {-1} {2}} end {align *}

The tricky thing is that no initial speed is given. But only starting position $$t = 0$$, and then instead a relation to the solution is given at 2 different times.

`NDSolve` Complain with this dreaded mistake

The boundary condition is not indicated on a single edge of the border
the computational domain.

And I do not know how to get rid of it. Here is the code

``````Delete everything[u, x, t];
pde = -D[u[x, t], {t, 2}]+[x, t] ==
D[u[x, t], {x, 2}]+ 2 * exp[-t]* (x - (1/2) * x ^ 2 + (1/2) * t - 1);

bc = {u[0, t] == 0, derivative[1, 0][u][1, t]    == 0};

ic = {u[x, 0] == x ^ 2 - 2 * x,
u[x, 1] == u[x, 1/2] + ((1/2) * x ^ 2 - x) * Exp[-1] - ((3 * x ^ 2) / 4 - (3/2) * x) * Exp[-2^(-1)]};

sol = NDSolve[{pde, ic, bc}, u, {x, 0, 1}, {t, 0, 1}]
``````

Here's the Maple code and the analytic solution it contains

``````pde: = -diff (u (x, t), t, t) + u (x, t) =
diff (u (x, t), x, x) + 2 * exp (-t) * (x- (1/2) * x 2 + (1/2) * t-1);
ic: = u (x, 0) = x ^ 2-2 * x,
u (x, 1) = u (x, 1/2) + (1/2) * x ^ 2-x) * exp (-1) - (3/4 * (x ^ 2) -3/2 * x) * exp (-1/2);
bc: = u (0, t) = 0, eval (diff (u (x, t), x), {x = 1}) = 0;
pdsolve ([pde, ic, bc], u (x, t))
``````

$$u (x, t) = – frac {e ^ {- t}} {2} (x ^ 2-2 x) (t-2)$$

Here is an animation of the Maple solution that I wanted to review

``````mapleSol[x_, t_] : = - (Exp[-t]/ 2) (x ^ 2 - 2 x) (t - 2)
Manipulate[
plot[mapleSol[mapleSol[mapleSol[mapleSol[x, t], {x, 0, 1}, PlotRange -> {{0, 1}, {-1, .1}}].
{{t, 0, "time"}, 0, 10, .1}
]
``````

Any suggestion how to fix the bug of NDSolve?

Using V 12 on Windows 10. ps. I also solved this by hand, but I can not find a Maple solution, and my solution looks wrong. I still have to find out why.

## Webhosting – risk of VPS – do I share information with other VPS users, such as IP? Can I find out I'm using a VPS solution?

I would like to implement a VPS for my company:

1. When surfing and using sites like Ebay / Facebook, can I find I'm using a VPS?
2. What information do I generally share while browsing next to IP?
3. When using a VPS, is it guaranteed that my IP address is unique? I mean, there are so many other customers using the same VPS company, and maybe there could be a problem.

## Why is the browser displaying my simple HTML source?

Hello friends,

Why is the browser displaying my simple HTML source?

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## Equation Solution – Definition of BCs for complex shape areas

is it possible that we can do such BCs for complex form with NDsolve,

The initial value for a disk is 1. Then we have to define a circle with such Dirichlet BCs: i.

code

``````(* Initial parameters *) needs["NDSolve`FEM`"];
Kappa = 1 / 10.0;
xmin = -1.0;
xmax = 1;
ymin = -1;
ymax = 1.0;

[CapitalOmega] = Rectangle[{xmin, ymin}, {xmax, ymax}];
RegionPlot[[[[[CapitalOmega], AspectRatio -> Automatic]mesh = ToElementMesh[[[[[CapitalOmega], "MaxCellMeasure" -> 1/1000];
mesh["Wireframe"]
n = length[mesh["Coordinates"]]u0 = ElementMeshInterpolation[{mesh}, RandomReal[{1, 1}, n]];
Plot3D[u0[x, y], {x, y} [Element] mesh][CapitalOmega]1 = hard disk[{0, 0}, xmax/10];
RegionPlot[[[[[CapitalOmega]1, aspect ratio -> automatic]mesh1 = ToElementMesh[[[[[CapitalOmega]1, "MaxCellMeasure" -> 1/1000];
mesh1["Wireframe"]
(* n = length[mesh1["Coordinates"]]u01 = ElementMeshInterpolation[{mesh1},0.5-RandomReal[{0,1},n]];
Plot3D[u01[x,y], {x, y} [Element]mesh1]*)

u = NDSolveValue[{d[x, y] - 1. -
4 * Kappa * Kappa * Laplace[d[x, y], {x, y}]== 0,
DirichletZustand[d[x, y] == 0, [CapitalOmega]1]d[x, y] == 0},
d, {x, y} [Element] [CapitalOmega]];
``````

## Equation Solution – Should Reduce specify all cases where \$ sqrt {x y} = sqrt x sqrt y \$?

My understanding is that `To reduce` Specifies all conditions (with or) where the input is true.

Now, $$sqrt {xy} = sqrt x sqrt y$$, from where $$x, y$$ are realunder the following three conditions / cases

begin {align *} x geq 0, y geq0 \ x geq0, y leq0 \ x leq0, y geq 0 \ end {align *}

but not when $$x <0, y <0$$

This is verified by this

``````Delete everything[x,y]
Provided[Element[{x,y},Reals]&& x> = 0 && y <= 0, simplify[ Sqrt[x*y] - sqrt[x]* Sqrt[y]]]Provided[Element[{x,y},Reals]&& x<= 0&&y>= 0, simplify[ Sqrt[x*y] - sqrt[x]* Sqrt[y]]]Provided[Element[{x,y},Reals]&& x<= 0&&y>= 0, simplify[ Sqrt[x*y] - sqrt[x]* Sqrt[y]]]Provided[Element[{x,y},Reals]&& x <= 0 && y <= 0, simplify[ Sqrt[x*y] - sqrt[x]* Sqrt[y]]]
``````

Why then?

``````    To reduce[ Sqrt[x*y] - sqrt[x]* Sqrt[y]== 0, {x, y}, reales]
``````

Do you just indicate one of the three cases mentioned above?

Is my understanding of `To reduce` wrong or should `To reduce` did the other two cases exist?

V 12 under Windows.

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## What is solution and code for this problem

If we have the string s1 and s2, we need to convert s1 to palindrome so that s1 contains s2 as a substring
In the minimal operation we can convert every character into another character. print -1 if not possible.

Ex.: S1 = arbcd, s2 = ar
Edition: 2
s1 = aaaaa, s2 = bbb
Edition: 3

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## How much solution to fill Paterson Universal Tank during film development?

Moving with Paterson tanks is done by rotating the roller, so only the film needs to be covered. For other systems where stirring may be by inversion or otherwise, the tank must be present nearly full (some air is needed).

The necessary chemical reactions are concentration- and time-dependent. More or less solution in the tank does not affect the results, as long as the film is sufficiently covered.