**> restart: with(plots): plotsetup(default): with(plottools): with(DEtools): **

**> Digits:=100: interface(displayprecision=3):**

> **#**Question:** **Does the 3-D system {xd=0,cd=0,qd=0} have a solution in (x,c,q) such that x>0, c>0 and 0.02<0.03 ?

**> f := x-> x^.5-.1*x:**

xd := f(x)-c;

cd := (diff(f(x),x)-q)*c;

qd := -(q-.2e-1)*(q-.3e-1)+(q-.25e-1)*(diff(f(x),x)-q)*c/(f(x)-c);

**> Digits:=10: interface(displayprecision=10):**

ss:= fsolve({xd,cd/c,qd}, {x,c,q}, avoid={x=0, c=0, q=0});

xs:=subs(ss,x): cs:=subs(ss,c): qs:=subs(ss,q):

> **#**Question: Is the solution Maple's fsolve return valid, precise, unique ? The system has an indeterminate form 0/0, which makes it difficult to check the alleged solution directly:

**> eval(xd,{x=xs,c=cs,q=qs}); eval(cd,{x=xs,c=cs,q=qs}); eval(qd,{x=xs,c=cs,q=qs});**

> **#**Let's visualize the 3-D system:

**> plotopts:=style=patchcontour, shading=none,lightmodel=light3,axes=boxed:**

px:=implicitplot3d({xd}, x=0..25,c=0..5,q=0.02..0.05, numpoints=10000, plotopts, colour=brown): pc:=implicitplot3d({cd/c}, x=0..25,c=0..5,q=0.02..0.05, numpoints=10000, plotopts, colour=green):

pq:=implicitplot3d({qd}, x=0..25,c=0..5,q=0.02..0.05, numpoints=10000, plotopts, colour=blue):

display3d({pq}, orientation=[-25,70]);

> **#**The plot is ragged, probably because of the 0/0 form. If I multiply by xd by (f(x)-c) and plot, I eliminate some of the raggedness and what looks like a vertical plane caused by the 0/0 form.

**> pq2:=implicitplot3d({qd*((x^.5-.1*x-c))}, x=0..25,c=0..5,q=0.02..0.05, numpoints=10000, plotopts, colour=black):**

display3d({pq,pq2}, orientation=[155,35]);

> **#**It looks like ploting xd*(f(x)-c) goes some way towards making the plot cleaner without eliminating (too much or any) information.

> **#**I plot the 3-D system together with the alleged solution.

**> points:=[[xs,cs,qs]]:**

**> ps:=pointplot3d(points,colour=yellow,symbol=circle,symbolsize=50):**

**> display3d({px,pc,pq,ps}, orientation=[-60,-100]);**

display3d({px,pc,pq2,ps}, orientation=[-60,-100]);

> **#**Based on the above 3-D plots, Maple's fsolve solution looks like what I'm looking for. But I can't be sure, because, frankly, the plot is not all that clean. Choosing a different orientation introduces some doubt into my mind.

**> display3d({px,pc,pq2,ps}, orientation=[170,95]);**

**> **

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