Simpler:

dsolve(de,implicit):
collect(%, x);
                   3
                  x           2
               - ---- + (-y(x)  + 2) x + _C1 + y(x) = 0
                  3

For the term in question, converted to exp form:

with(MultiSeries):assume(S0,'real',S0<1,S0>0);assume(R3,'real');
assume(R1,'real');assume(variphi,'real');assume(l,'real');

(1-R1)*(variphi*(Sigma0+Sigma1/l)*l+R3*l*coth(l*S0))*(cosh(l*S0)*
sinh(l)-cosh(l)*sinh(l*S0))*coth(l*S0)*1/sinh(l)/
(R1*R2*(sinh(l*S0)-cosh(l*S0)*coth(l))-coth(l*S0)*cosh(l*S0)+
cosh(l*S0)*coth(l)):

f:=convert(%,exp):

I get one expansion for the range 0< S0 <1/3, and 1/3< S0 <1/2 :

asympt(f,l,3) assuming additionally, S0 < 1/3:
map(simplify,%);

  (-R3 + variphi Sigma0 R1 - variphi Sigma0 + R3 R1) l
  ----------------------------------------------------
                       R1 R2 + 1

           variphi Sigma1 (-1 + R1)
         + ------------------------ + O(l) exp(-2 S0 l)
                  R1 R2 + 1


asympt(f,l,3) assuming additionally, 1/3<S0,S0<1/2:
map(simplify,%);


  (-R3 + variphi Sigma0 R1 - variphi Sigma0 + R3 R1) l
  ----------------------------------------------------
                       R1 R2 + 1

           variphi Sigma1 (-1 + R1)
         + ------------------------ + O(l) exp(-2 S0 l)
                  R1 R2 + 1

And a different third term above 1/2:

asympt(f,l,3) assuming additionally, 1/2<S0,S0<2/3:
map(simplify,%);

  (-R3 + variphi Sigma0 R1 - variphi Sigma0 + R3 R1) l
  ----------------------------------------------------
                       R1 R2 + 1

           variphi Sigma1 (-1 + R1)
         + ------------------------ + O(l) exp(2 (-1 + S0) l)
                  R1 R2 + 1

asympt(f,l,3) assuming additionally, 2/3<S0,S0<1:
map(simplify,%);

  (-R3 + variphi Sigma0 R1 - variphi Sigma0 + R3 R1) l
  ----------------------------------------------------
                       R1 R2 + 1

           variphi Sigma1 (-1 + R1)
         + ------------------------ + O(l) exp(2 (-1 + S0) l)
                  R1 R2 + 1

So, if the expansion does depend on S0, it seems like the real critical value is 1/2.

Then you will need to identify the critical values of S0 for which a pair of exponents become equal. I.e. the sequence of exponents produced by the error messages like:

Error, (in MultiSeries:-multiseries) unable to sort exponents, 
{-6, -4, -2, -2/S0, -1/S0*(2*S0+2), -1/S0*(4*S0+2)}

or

Error, (in assuming) when calling 'MultiSeries:-multiseries'. 
Received: 'unable to sort exponents, {-2/S0, -6/S0*S0, -4/S0*S0, 
-2/S0*S0, -1/S0*(2*S0+2), -1/S0*(4*S0+2)}, MultiSeries:-multiseries'

where I have added assuming additionally, S0<1/2 to the command.

This means, take every pair of exponents, equate them and solve for S0. Assuming that S0 is within each interval delimited by these critical values, the asymtotic expansion should work and be expressed symbolically in terms of S0. And for the critical values, the expansion should be obtained separated.

You could do something like

\MapleInput{x\^{}2;}

You could do something like

\MapleInput{x\^{}2;}

The Mathematica approach to this subject, as described briefly here, tells about its frontend:

In addition, Mathematica's notebook front end is a separate process from its computational kernel, allowing them to run on separate cores or CPUs. This gives a responsive interface even when the kernel core is under full load.

Can Maple's interface execute in a  separate core? This might be relevant because of the performance problems of the Standard GUI.

To pull out symbolic factors I would convert to a Heaviside representation like e.g.:

j:=piecewise(x<0,-A*x^2,x>0,A*x):
j1:=(factor@convert)(j,Heaviside);
            j1 := A x (-x + x Heaviside(x) + Heaviside(x))

evalf(IntegrationTools:-Expand(Int(j1,x=-1..1)));
                            0.1666666666 A

It seems like the default normalization of piecewise is with factors "in" as here:

convert(j1,piecewise);
                      {        2
                      {    -A x            x < 0
                      {
                      { A undefined        x = 0
                      {
                      {     A x            0 < x

And probably there is no command implemented to factor them out.

To pull out symbolic factors I would convert to a Heaviside representation like e.g.:

j:=piecewise(x<0,-A*x^2,x>0,A*x):
j1:=(factor@convert)(j,Heaviside);
            j1 := A x (-x + x Heaviside(x) + Heaviside(x))

evalf(IntegrationTools:-Expand(Int(j1,x=-1..1)));
                            0.1666666666 A

It seems like the default normalization of piecewise is with factors "in" as here:

convert(j1,piecewise);
                      {        2
                      {    -A x            x < 0
                      {
                      { A undefined        x = 0
                      {
                      {     A x            0 < x

And probably there is no command implemented to factor them out.

Indeed, crosses and dots can be used to represent a vector normal to a plane. But in 3D  a vector along the 4th cartesian component is normal to all of the three cartesian orthogonal planes (where one of the coordinates is set to a constant), and it could be based at any point within the plot volume. Besides, you would need a representation for any vector with both parallel (3D) and normal components.

And note that what you observe is a projection of that volume onto the screen or the printout paper. So, you would need something that is a consistent generalization of the cross and dot scheme and at the same time does not produce a completely confusing result when projected.

In any case, Maple does not have any command for that. I am not even sure whether Maple is the best tool for hacking something like that.

Indeed, crosses and dots can be used to represent a vector normal to a plane. But in 3D  a vector along the 4th cartesian component is normal to all of the three cartesian orthogonal planes (where one of the coordinates is set to a constant), and it could be based at any point within the plot volume. Besides, you would need a representation for any vector with both parallel (3D) and normal components.

And note that what you observe is a projection of that volume onto the screen or the printout paper. So, you would need something that is a consistent generalization of the cross and dot scheme and at the same time does not produce a completely confusing result when projected.

In any case, Maple does not have any command for that. I am not even sure whether Maple is the best tool for hacking something like that.

If you do not provide your exact input, it is impossible to know what Maple understood from that, or fails to do.

If you do not provide your exact input, it is impossible to know what Maple understood from that, or fails to do.

In a 3D field plot, you are missing one coordinate plus one vector component.

Say that somewhere the field points along the suppressed dimension, so no arrow would be ploted in the other three dimensions plot, or at most a point., as if the field were vanishing. So, from a sequence of such 3D plots for different values of the 4th coodinate you would not be able to decide which is the case.

In a 3D field plot, you are missing one coordinate plus one vector component.

Say that somewhere the field points along the suppressed dimension, so no arrow would be ploted in the other three dimensions plot, or at most a point., as if the field were vanishing. So, from a sequence of such 3D plots for different values of the 4th coodinate you would not be able to decide which is the case.

How would you visualize a four-dimensional field (assuming F(t)=0)?