nm

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10 years, 205 days

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These are answers submitted by nm

if you use mul instead of product you could use, for specific the following

restart;
p:=(x,n)->expand(mul( (x+i), i=1..n)):
p(x,5);
coeff( %,x^2)

 

p(x,10);
coeff( %,x^2)

if you use product then it does not seem possible since specific value of n is not known:

restart;
p:=(x,n)->product( (x+i), i=1..n);
p(x,n)

Newton's law of cooling is    q'(t) + k*( q(t) - 25) = 0.

I do not know why you have "t" in the right side. 

I am will solve it using 0 on right side.  You can always change it if you want, but that is not the Newton law of cooling I know.  Why do you have time on the right side? Is there a heating source in the room that generates heat that increases with time  linearly? This is not physically possible.

restart;
k:=2;
ode:=diff(q(t),t)+k*(q(t)-25)=0;
ic:=q(0)=100;
sol:=dsolve([ode,ic]);
#temp after 30 minutes.
evalf(eval(rhs(sol),t=30));
plot(rhs(sol),t=0..30, view=[0..30,1..100]);

So it will at room temp. After 30 minutes. You have very fast cooling there due to very large k. Normally k is a very small value..

Here is the plot of body temp. vs. time

To find the time to reach 50 degrees, solve the equation:

#find how long it takes to go to 50;
eq:=50=rhs(sol);
evalf(solve(eq,t));

0.5493061445

A little over 1/2 second.

You can repeate the same if you want to have "t" on the RHS of the ode.

 

Here is a version with upper case E for the fortran notation

When running the Maple code at the end, it will generate this Latex code

\documentclass{article}
\usepackage{amsmath}
\usepackage{multirow}
\renewcommand{\arraystretch}{1.2}
\usepackage{color, colortbl}
\definecolor{Gray}{gray}{0.9}
\begin{document}
\begin{table}[hbt!]
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
\rowcolor{Gray} $x$&$t$&$f \! \left(x , t\right)$&$g \! \left(x , t\right)$\\ \hline\hline 
$ 0.125$&$ 0.125$&$ 0.015625$&$ 1.953\, \mathrm{E}^{-03}$ \\ \hline 
$ 0.125$&$ 0.375$&$ 0.046875$&$ 5.859\, \mathrm{E}^{-03}$ \\ \hline 
$ 0.125$&$ 0.625$&$ 0.078125$&$ 9.766\, \mathrm{E}^{-03}$ \\ \hline 
$ 0.125$&$ 0.875$&$ 0.109375$&$ 1.367\, \mathrm{E}^{-02}$ \\ \hline 
$ 0.375$&$ 0.125$&$ 0.046875$&$ 1.758\, \mathrm{E}^{-02}$ \\ \hline 
$ 0.375$&$ 0.375$&$ 0.140625$&$ 5.273\, \mathrm{E}^{-02}$ \\ \hline 
$ 0.375$&$ 0.625$&$ 0.234375$&$ 8.789\, \mathrm{E}^{-02}$ \\ \hline 
$ 0.375$&$ 0.875$&$ 0.328125$&$ 1.230\, \mathrm{E}^{-01}$ \\ \hline 
$ 0.625$&$ 0.125$&$ 0.078125$&$ 4.883\, \mathrm{E}^{-02}$ \\ \hline 
$ 0.625$&$ 0.375$&$ 0.234375$&$ 1.465\, \mathrm{E}^{-01}$ \\ \hline 
$ 0.625$&$ 0.625$&$ 0.390625$&$ 2.441\, \mathrm{E}^{-01}$ \\ \hline 
$ 0.625$&$ 0.875$&$ 0.546875$&$ 3.418\, \mathrm{E}^{-01}$ \\ \hline 
$ 0.875$&$ 0.125$&$ 0.109375$&$ 9.570\, \mathrm{E}^{-02}$ \\ \hline 
$ 0.875$&$ 0.375$&$ 0.328125$&$ 2.871\, \mathrm{E}^{-01}$ \\ \hline 
$ 0.875$&$ 0.625$&$ 0.546875$&$ 4.785\, \mathrm{E}^{-01}$ \\ \hline 
$ 0.875$&$ 0.875$&$ 0.765625$&$ 6.699\, \mathrm{E}^{-01}$ \\ \hline 
\end{tabular}
\end{center}
\end{table}
\end{document}

Which you can compile using your favourite Latex compiler. The Maple code uses 

I kept the exponent of E as string, so it shows as "-03" instead of -3. To have the same size for all numbers. You can easily change it if you want -3 instead.


 

restart:
interface(rtablesize=20):
f:=(x,t)->x*t;
g:=(x,t)->x^2*t;

B:=Matrix([seq(seq([i,j,f(i,j),g(i,j)],j=0.125..0.875, 0.25),i=0.125..0.875,0.25)]);

convert_it:=proc(matrix_entry)
  local res;
  res:= nprintf(`#mn(\"%1.3e\");`,matrix_entry);
  res:= convert~(res,string);
  res:= StringTools:-Delete(res,1..5);
  res:= StringTools:-Delete(res,-3..-1);
  res:= StringTools:-Split(res,"e");
  return parse(res[1]),res[2]
end proc:

toX:=proc(x)
  Latex(x,output=string);
end proc:

Lat:=proc(M::Matrix,header::list)::string;
  local s::string:="";
  local nRows::posint;
  local nCols::posint;
  local N,item;
  local i,j;
  local the_number,the_exponent;
  
  nRows,nCols:=LinearAlgebra:-Dimension(M);

  s:=cat("\\documentclass{article}
\\usepackage{amsmath}
\\usepackage{multirow}
\\renewcommand{\\arraystretch}{1.2}
\\usepackage{color, colortbl}
\\definecolor{Gray}{gray}{0.9}
\\begin{document}
\\begin{table}[hbt!]
\\begin{center}
\\begin{tabular}{|",seq("c|",i=1..nCols),"}\\hline
\\rowcolor{Gray} ");
    for N,item in header do        
        if N<numelems(header) then
           s:=cat(s,"$",toX(parse(item)),"$&");
        else
           s:=cat(s,"$",toX(parse(item)),"$\\\\ \\hline\\hline \n");
        fi;
    od;
    
    for i from 1 to nRows do
        for j from 1 to nCols do
            if j=nCols then
               the_number,the_exponent:=convert_it(M[i,j]);
               s:=cat(s,"$",toX(the_number),"\\, \\mathrm{E}^{",the_exponent,"}$ \\\\ \\hline \n");
            else
               s:=cat(s,"$",toX(M[i,j]),"$&");
            fi;        
        od;
    od;
    
    s:=cat(s,"\\end{tabular}
\\end{center}
\\end{table}
\\end{document}
");
    return s;        

  
 end proc:
  

f := proc (x, t) options operator, arrow; x*t end proc

g := proc (x, t) options operator, arrow; x^2*t end proc

Matrix(%id = 36893490297603924324)

s:=Lat(B,[x,t,`f(x,t)`,`g(x,t)`]):
printf("%s",s)

\documentclass{article}
\usepackage{amsmath}
\usepackage{multirow}
\renewcommand{\arraystretch}{1.2}
\usepackage{color, colortbl}
\definecolor{Gray}{gray}{0.9}
\begin{document}
\begin{table}[hbt!]
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
\rowcolor{Gray} $x$&$t$&$f \! \left(x , t\right)$&$g \! \left(x , t\right)$\\ \hline\hline
$ 0.125$&$ 0.125$&$ 0.015625$&$ 1.953\, \mathrm{E}^{-03}$ \\ \hline
$ 0.125$&$ 0.375$&$ 0.046875$&$ 5.859\, \mathrm{E}^{-03}$ \\ \hline
$ 0.125$&$ 0.625$&$ 0.078125$&$ 9.766\, \mathrm{E}^{-03}$ \\ \hline
$ 0.125$&$ 0.875$&$ 0.109375$&$ 1.367\, \mathrm{E}^{-02}$ \\ \hline
$ 0.375$&$ 0.125$&$ 0.046875$&$ 1.758\, \mathrm{E}^{-02}$ \\ \hline
$ 0.375$&$ 0.375$&$ 0.140625$&$ 5.273\, \mathrm{E}^{-02}$ \\ \hline
$ 0.375$&$ 0.625$&$ 0.234375$&$ 8.789\, \mathrm{E}^{-02}$ \\ \hline
$ 0.375$&$ 0.875$&$ 0.328125$&$ 1.230\, \mathrm{E}^{-01}$ \\ \hline
$ 0.625$&$ 0.125$&$ 0.078125$&$ 4.883\, \mathrm{E}^{-02}$ \\ \hline
$ 0.625$&$ 0.375$&$ 0.234375$&$ 1.465\, \mathrm{E}^{-01}$ \\ \hline
$ 0.625$&$ 0.625$&$ 0.390625$&$ 2.441\, \mathrm{E}^{-01}$ \\ \hline
$ 0.625$&$ 0.875$&$ 0.546875$&$ 3.418\, \mathrm{E}^{-01}$ \\ \hline
$ 0.875$&$ 0.125$&$ 0.109375$&$ 9.570\, \mathrm{E}^{-02}$ \\ \hline
$ 0.875$&$ 0.375$&$ 0.328125$&$ 2.871\, \mathrm{E}^{-01}$ \\ \hline
$ 0.875$&$ 0.625$&$ 0.546875$&$ 4.785\, \mathrm{E}^{-01}$ \\ \hline
$ 0.875$&$ 0.875$&$ 0.765625$&$ 6.699\, \mathrm{E}^{-01}$ \\ \hline
\end{tabular}
\end{center}
\end{table}
\end{document}

 


 

Download code.mw

 

restart;
pde1:= diff(C(x,t),t)=d* diff(C(x,t),x$2)- k*epsilon(x)*C(x,t);
pde2:= diff(epsilon(x),x)=-mu*k*epsilon(x)*C(x,t);

pdsolve([pde1,pde2],[C(x,t),epsilon(x)]) assuming x>a,x<=b,t>0

....  Other long and complicated solutions are also given in addition to the above, in terms or RootOf (ps. I renamed your "D2" to "d"

 

Maple 2020.2

On windows:

Click Start->  Then type in the search box "environment" and select the tiop option. This will bring up "System properties". Click on botton called "Environemnt variables" . In the top window, for "User Variables", click "NEW" and add  Variable caled TEXINPUTS and give it value of the full path of the folder where you your Maple style files are.  On windows it is in

"C:\Program Files\Maple 2020\etc"

and click OK. And start  exnicCenter and compile, now it will find it. Make sure in your Latex file you include it as follows (if it is not allready there)

    \usepackage{maplestd2e}

Now MikTex compiler, when invoked by TexnicCenter will lookup this environment variable and use it to search for maple latex style files.

There is another method to do this, Using MIKTEX Console and then going to settings->Directories and adding the above Maple folder path to the TEXMF folders listed there allready.  

I find editing the Environemnt variables and adding TEXINPUTS easier. 

 

b:=(X__2-I__22)/((x__1-I__11)*(X__2-I__22)-I__12*I__21);

You have a typo there. It should be X__1 and not lower case x__1 

simplify(a) almost gets there:

But to get to exactly b, which is 

I do not know, (could not find direct way, but may be there is). other than looking at it, and do the following

restart;
a:=-(I__22-X__2)/(I__11*I__22-I__11*X__2-I__12*I__21-I__22*X__1+X__1*X__2);
a:=simplify(a);
term:=X__1 - I__11;
algsubs(term=A,a);
collect(%,A):
subs(A=term,%);

Getting CAS to give the same exact form one wants, is one of the hardest things. sometimes CAS has it own way of formating expressions.   So some manual steps would be needed.

 

 

You could check the difference is zero between each corresponding elements?

V1 := Vector[column](8, [1, 2, 2, 1, 3, A, B, 1/(A + B)^2]);
V2 := Vector[column](8, [1, 2, 2, 1, 3, A, B, 1/(A^2 + 2*A*B + B^2)]);
ArrayTools:-IsZero(simplify(V1-V2))

      true

 

I generate everything from Maple using Latex.  CAS systems (either Maple or Mathematica) can't compete with Latex when it comes to formating output. 

ps. I get different numbers for f(x,t) and g(x,t). May be you used different definition than you showed