Why did you say that wasn't what you wanted?  That was it exactly.  Or you could use implicitdiff to get an equivalent result.  Then replace diff(y(x),x) with -1/diff(y(x),x) to get a differential equation for the orthogonal trajectories.

You made no assumption on x.  Basically I think the problem that Maple needs to worry about is that the antiderivative may have a discontinuity (see e.g. the thread  Maple Integration Error).  Certainly there is likely to be trouble if the integrand itself went to infinity, which could happen if x was imaginary.

I'm not sure what you mean by

"Instead of just outputting the matrix for aesthetics, how can i then pass those values on to a matrix P?"

My procedure matrixMaplet returns the Matrix, and you can do whatever you wish with the result, for example assign it to P:

P := matrixMaplet(3,4);

A useful addition to matrixMaplet might be an optional third argument that provides initial values for the entries.  Thus:

matrixMaplet:= proc(m::posint, n::posint, 
   M::{Matrix,matrix,listlist})
     local maplet, M0;
     uses Maplets[Elements];
     if nargs = 3 then M0 := M
     else M0 := Matrix(m,n)
     end if;   
     maplet:= Maplet([
         ["Edit the matrix, please"],
         seq([seq(TextBox[_entry||i||j]('width' = 5,
              'value'=M0[i,j]),j = 1..n)],
             i = 1..m),
         [Button("Done",
           Shutdown([seq(seq(_entry||i||j,j=1..n),i=1..m)]))]
        ]);
   Matrix(m,n,map(parse,Maplets[Display](maplet)));
  end proc:

Q:= <<1,2>|<3,4>|<5,6>>;
Q:= matrixMaplet(2, 3, Q);

I'm not sure what you mean by

"Instead of just outputting the matrix for aesthetics, how can i then pass those values on to a matrix P?"

My procedure matrixMaplet returns the Matrix, and you can do whatever you wish with the result, for example assign it to P:

P := matrixMaplet(3,4);

A useful addition to matrixMaplet might be an optional third argument that provides initial values for the entries.  Thus:

matrixMaplet:= proc(m::posint, n::posint, 
   M::{Matrix,matrix,listlist})
     local maplet, M0;
     uses Maplets[Elements];
     if nargs = 3 then M0 := M
     else M0 := Matrix(m,n)
     end if;   
     maplet:= Maplet([
         ["Edit the matrix, please"],
         seq([seq(TextBox[_entry||i||j]('width' = 5,
              'value'=M0[i,j]),j = 1..n)],
             i = 1..m),
         [Button("Done",
           Shutdown([seq(seq(_entry||i||j,j=1..n),i=1..m)]))]
        ]);
   Matrix(m,n,map(parse,Maplets[Display](maplet)));
  end proc:

Q:= <<1,2>|<3,4>|<5,6>>;
Q:= matrixMaplet(2, 3, Q);

> f:= x/(x^2+y^2+z^2)^(3/2);
   F1:= int(f,z=-b..b) assuming positive;

F1 := 2/(x^2+y^2+b^2)^(1/2)*b/(x^2+y^2)*x

So far so good

> Q:=int(F1,y=c..a+c) assuming positive;

Q := -I*arctanh((b^2+c*x*I+x^2)/b/(b^2+c^2+x^2)^(1/2))-I*arctanh((-b^2+c*x*I-x^2)/b/(b^2+c^2+x^2)^(1/2))+arctanh((b^2+a*x*I+c*x*I+x^2)/b/(2*a*c+c^2+x^2+a^2+b^2)^(1/2))*I+arctanh((-b^2+a*x*I+c*x*I-x^2)/b/(2*a*c+c^2+x^2+a^2+b^2)^(1/2))*I

It's curious that Maple finds an integral that appears complex, even though (under the assumption that x, a, b, c are all positive) the integral should obviously be real.

> F2:= simplify(evalc(int(F1,y))) assuming positive;

F2 := -arctan(x*y/(x^2+b^2+b*(x^2+y^2+b^2)^(1/2)))+1/2*arctan(-x*y,b*(x^2+y^2+b^2)^(1/2)-x^2-b^2)-1/2*arctan(x*y,b*(x^2+y^2+b^2)^(1/2)-x^2-b^2)

> Q := eval(F2,y=a+c) - eval(F2, y=c);

Q := -arctan(x*(a+c)/(x^2+b^2+b*(x^2+(a+c)^2+b^2)^(1/2)))+1/2*arctan(-x*(a+c),b*(x^2+(a+c)^2+b^2)^(1/2)-x^2-b^2)-1/2*arctan(x*(a+c),b*(x^2+(a+c)^2+b^2)^(1/2)-x^2-b^2)+arctan(x*c/(x^2+b^2+b*(b^2+c^2+x^2)^(1/2)))-1/2*arctan(-x*c,b*(b^2+c^2+x^2)^(1/2)-x^2-b^2)+1/2*arctan(x*c,b*(b^2+c^2+x^2)^(1/2)-x^2-b^2)

I have no trouble using display with several contourplot3d's.  It's hard to tell what's gone wrong from this distance.  Can you upload your worksheet?

What I wrote is a solution of the equation as you wrote it: D*diff(c(x,t),t)=diff(c(x,t),x,x).
I think it's more common to write the equation as diff(c(x,t),t)=D*diff(c(x,t),x,x), and that would explain the formulas you see in Crank.

I don't know if there is a closed-form solution for the case where the diffusion constant is different in the two materials. 

What I wrote is a solution of the equation as you wrote it: D*diff(c(x,t),t)=diff(c(x,t),x,x).
I think it's more common to write the equation as diff(c(x,t),t)=D*diff(c(x,t),x,x), and that would explain the formulas you see in Crank.

I don't know if there is a closed-form solution for the case where the diffusion constant is different in the two materials. 

Well, you could do something like this with convertsys:

> de := diff(y(t), t$2) + diff(y(t),t)+y(t) = 0:
   convertsys({de}, [], y(t), t, Y,YP);
   sys:= eval(%[1],[YP=diff([y(t),v(t)],t),Y=[y(t),v(t)]]);

But that's no better than doing it on your own without convertsys:

> sys := [diff(y(t), t) = v(t), 
    isolate(subs(diff(y(t),t)=v(t), de),diff(v(t),t))];

Well, you could do something like this with convertsys:

> de := diff(y(t), t$2) + diff(y(t),t)+y(t) = 0:
   convertsys({de}, [], y(t), t, Y,YP);
   sys:= eval(%[1],[YP=diff([y(t),v(t)],t),Y=[y(t),v(t)]]);

But that's no better than doing it on your own without convertsys:

> sys := [diff(y(t), t) = v(t), 
    isolate(subs(diff(y(t),t)=v(t), de),diff(v(t),t))];

What is the particular LF in this case?  Do assumptions such as  t::real
make a difference (depending on what LF is, that might be necessary for convergence)?

Why?  Because that's the way the software was designed.  But you do have a point: this is a common enough situation that I think it would be worthwhile to have an option for phaseportrait to automatically transform a second-order DE to a system. 

Why?  Because that's the way the software was designed.  But you do have a point: this is a common enough situation that I think it would be worthwhile to have an option for phaseportrait to automatically transform a second-order DE to a system. 

I've never encountered this word (which doesn't prove anything, but I've been a mathematician for a long time).  Oxford English Dictionary, dictionary.com, wikipedia, MathWorld, and Maple's own math dictionary have nothing on it.  On the other hand, google does turn up about 375 hits, including quite a number of research papers.  Of those I looked at, all had Chinese authors. 

I've never encountered this word (which doesn't prove anything, but I've been a mathematician for a long time).  Oxford English Dictionary, dictionary.com, wikipedia, MathWorld, and Maple's own math dictionary have nothing on it.  On the other hand, google does turn up about 375 hits, including quite a number of research papers.  Of those I looked at, all had Chinese authors.