## How can I know what is # in equation?

I am trying to find the solution by using Reduce to fit my data with FindFit. And I don’t know what # in the result equation. My code is:

Reduce[kh == (oh (0.2 - 2 (0.5 (ohmax - oh))^2))/(    ph2o*po2^(1/2) ko (0.5 (ohmax - oh))^2) && kh > 0 && ph2o > 0 &&    po2 > 0 && ko > 0 && oh > 0, oh, Reals] 

And I get this result:

I need to put the result equation into the FindFit however, I don’t know what # means and how I should modify the equation so that I won’t have any issue with FindFit.

Thank you.

## How to Solve Functional Differential Equation

I tried to solve this equation numerically.

eqnEx = x''[t] + x[2 t] == 0; NDSolve[{eqnEx, x[1] == 10, x'[1] == 0}, x[t], {t, 1, 10}] 

Two important character of the equation are that function variable is 2t and initial condition is given at t=1 .

After Calculation, Mathematica gives this error message

NDSolve : The method currently implemented for delay differential equations does not support delays that depend directly on the time variable or dependent variables 

After google, I find these kind of equation are called Functional Differential Equation and I could not obtain how to solve it. Is there a way to solve these kind of equation numerically?

## How to fit a curve in a picture with an equation?

For a curve taken from a picture, is there any method to fit it with an equation if it appears to be some standard curve?

For example, in the following picture, the curve looks like an ellipse or a circle or something else of a conic section. How can I fit the shape with a proper equation using Mathematica? Furthermore, is it possible to assess two different fittings with a criterion (e.g. error)? Thank you in advance.

## Solve an algebraic equation with an integral

I am trying to compute for the variable zm in terms of t which is written as an algebraic equation with an integral in it. The final answer should be zm = zm[t].

t - Integrate[(c[zm] z^(d - 1))/(f[z] Sqrt[f[z] + c[zm]^2 z^(2 d - 2)]), {z, 0, zm}] == 0

Just a note, c[zm] contains a negative sign inside the square root so that zm must be greater than zh in order for c[zm] to be real.

d = 3; zh = 2; c[zm_] := Sqrt[-(1 - zm^(d + 1)/zh^(d + 1))]/zm^(d - 1); f[z_] := (1 - z^(d + 1)/zh^(d + 1));  In[8]:= Integrate[(c[zm] z^(d - 1))/(f[z] Sqrt[f[z] + c[zm]^2 z^(2 d - 2)]), {z, 0, zm},   Assumptions -> zm > 2]  Out[8]= (1/64)*Pi*((-32 - 32*I) - (Sqrt[2*Pi]*zm*Sqrt[-16 + zm^4]*(-1 + Hypergeometric2F1[-(1/4), 1, 1/4, 16/zm^4]))/Gamma[5/4]^2)  Solve[t - Integrate[(c[zm] z^(d - 1))/(f[z] Sqrt[f[z] + c[zm]^2 z^(2 d - 2)]), {z, 0, zm}] == 0 , zm] 

I am not sure if using Solve can really find the expression, also the result of the integral contains an imaginary term, but it should not right since c[zm] is real from the Assumptions -> zm>2?

## Solve an equation in mathematica without replacing the value of the parameter

Suppose I want to solve a simple equation in mathematica x-a=0.So I am writing the mathematica code for this as below:

Solve[x - a == 0, x] 

So the output will be as below:

{{x -> a}} 

Now suppose I have assigned a value for ‘a’ beforehand and then want to solve the same equation.So my code will look like below:

a = 1; Solve[x - a == 0, x] 

And the output in this case will be like below:

{{x -> 1}} 

Now if I want an output in this case as {{x -> a}},what modification should I do in my code ?

Note: Clear[a] will work,but I don’t want to remove permanently the assigned value to a

## Von Neumann Equation Density Matrix Implementation

I’m trying to implement the von Neumann Equation for a given 4×4 density Matrix with a time dependent Hamiltonian Hp[t_] in Mathematica but I get stuck.

Format[y[a__]] := Subscript[y, a] rho[t_] := Array[x[##][t] &, {4, 4}]  sol = NDSolve[{I*rho'[t] == Hp[t].rho[t] - rho[t].Hp[t],     rho[0] == rhoIni}, {rho}, {t, 0, 10}] 

However I only get the output

{{rho -> rho}} 

So I guess something is structurally wrong with my code. I try to extract a solution by writing

rho[t_] = rho[t] /. sol 

But this doesn’t work as there is no solution anyways. Maybe you can help me

## How can I find in mathematica if my equation have solution or not on a given interval?

I’m new to mathematica, I usually used wolfram alpha, however since the equation that I’m working with is a long equation I need to use mathematica. This is an example of the problem. So, I want to know if the equation will have solution or not when m>=4 and n>=3.41421m. I don’t know what command to use and when I enter this, it said that m>=4 is not a valid variable. What should I do?

Solve[{Binomial[n-1,2]-2[(Binomial[m-1,2])+(n-m)(m-1)-1]<=0}, {m>=4, n>=3.41421m}, Reals]

## Does anyone here know how can we extract Lambda from the following equation

I am working on an estimation problem where I have the following

Code is

A = 1 + Pi*Csc[(1 + Pi)*\[Lambda]]^2*     (Pi + Sin[2*(1 + Pi)*\[Lambda]]) 

Can anyone help me how can we extract $$\lambda$$ from this equation in Mathematica?

## NSolve connot solve my equation

I’m trying to solve $$\left(\frac{hc}{k_B} – 5x\right)e^{hc/k_Bx} + 5x = 0,$$ where $$h = 6.626 \times 10^{-34}$$, $$k = 1.381 \times 10^{-23}$$ and $$c = 3 \times 10^8$$. My code is

h = 6.626*^-34; k = 1.381*^-23; c = 3*^8; NSolve[((h c)/k - 5 x) Exp[(h c)/k x] + 5 x == 0, x] 

but Mathematica doesn’t solve the equation, am I doing something wrong?

## finding FWHM of a dataset with unknown mathematical equation

I have a dataset. I have plotted using "Listloglinearplot". Now I need to find the FWHM (full width half maxima) of the same, However I dont know which mathematical eqution describes best to fit my dataset to find out FWHM. I have the following data and plot:

dataset={{0., 0.0518175}, {1., 0.0306299}, {1.9, 0.610295}, {2.,    1.32653}, {2.2, 4.01183}, {2.5, 6.37931}, {3., 6.50091}, {5.,    6.54052}, {6., 6.57276}, {8.2, 6.59119}, {15., 6.56125}, {20.,    6.5267}, {30., 6.4484}, {45., 6.2987}, {60., 6.11953}, {75.,    5.84962}, {90., 5.43738}, {100., 4.96757}, {105., 4.54382}, {120.,    3.42917}, {135., 2.23092}, {150., 1.55222}, {165., 0.679385}, {180.,    0.444479}} dataplot =   ListLogLinearPlot[dataset,    PlotStyle -> {Dashing[{.0071, 0.005, 0.005}], Blue},    PlotMarkers -> {\[FilledCircle], 15}, Frame -> True,    FrameStyle -> Directive[Black, Thickness[0.002]],    FrameLabel -> {Style["x", Black, FontFamily -> "Times New Roman",       FontSize -> 26],      Style["y", Black, FontFamily -> "Times", FontSize -> 26]},    PlotRange -> {{0, 190}, {1, 7.2}}, FrameTicks -> Automatic,    ImageSize -> 650,    BaseStyle -> {FontFamily -> "Times", FontSize -> 10}]