Question: Seeking assurances about an alleged solution to a 2D ODE BVP

This is a problem about which I have previously posted here on several occasions, please stop reading now if you are fed up with it, I do apologize.

Let me go straight to the somewhat-reduced-form problem. I have two (non-autonomous) ODEs in C(x) and Q(x)

ODE1 := diff(C(x),x) = s*(fx(x)-Q(x))*C(x) / (f(x)-C(x));
ODE2 := diff(Q(x),x) = (Q(x)-b1)*(b2-Q(x)) / (f(x)-C(x)) + (Q(x)-b)*s*(fx(x)-Q(x))*C(x) / (f(x)-C(x))^2;

In the above, f(x) is a known function and s, b1, b2, b are known real parameters. A particular parameterization is the following:

s:=1/2: b1:=0.02: b2:=0.03: b:=0.25:
ODE1 := diff(C(x),x) = 1/2*(.5/x^.5-.5e-1-Q(x))*C(x)/(x^.5-.5e-1*x-C(x));
ODE2 := diff(Q(x),x) = (Q(x)-.2e-1)*(.3e-1-Q(x))/(x^.5-.5e-1*x-C(x))+1/2*(Q(x)-.25e-1)

I want to know if there is a solution to this ODE system for a particular set of boundary conditions, namely for the following:

qs:=b: xs:=fsolve(fx(x)-b,x): cs:=f(xs): 

where the notation means qs = Q(xs) and cs = C(xs). Specifically, I have:

          ss := [44.4444444444444, 4.44444444444444, 0.025]

For reasons I have discussed elsewere (see references below) I have become convinced that there exists a solution to this ODE system for the boundary conditions thus specified but only for a range of the independent variable x, namely for x<xs, but not for x>xs. Does anyone agree/disagree with this assessment?

Maple's dsolve produces the following,

from "below" :

SYS := ODE1, ODE2: VAR := C(x), Q(x): INI1 := C(xs)=0.9999999*cs, Q(xs)=qs:
SOL1 := dsolve({SYS, INI1}, {VAR}, type=numeric, output=listprocedure, stiff=true, range=0.0001..xs);

from "above":

INI2 := C(xs)=1.0099*cs, Q(xs)=qs:
SOL2 := dsolve({SYS, INI2}, {VAR}, type=numeric, output=listprocedure, stiff=true, range=xs..2*xs):

The resulting solution is displayed below:



While Maple spits out a solution, it is quite clear that the portion of the solution for x>xs is not actually a solution. What is less clear is whether the portion for x<xs is a valid solution or not. I am inclined to think it is for reasons discussed earlier here and summarized in this worksheet. The question I was asking then, and am asking now again for the same re-formulated problem is:

Why do I think that the dsolve solution from above is not valid? because of what appears to be a jump at some value just above xs. Why do I think that the solution from below is valid? Well, I'm not certain, but an approximation detailed in the worksheet just mentioned seems to show it, and from below we do not observe a jump, instead we observe that the slope of Q(x) appears to tend to infinity as x approaches xs from below.

If my summary of the problem is unclear, I apologize and will try to reformulate it. If there are missing pieces of information, please do let me know and I will supply them.

I thank you all for your help.

The calculations summarized above are contained in the following worksheet:

Further references to my problem:



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