I was bothered by the fact that the simple answer, b=11, was obtained after combining some considerably large and messy expressions.  I looked and found a very simple solution that avoids the mess.  Maple is not of much help in this alternative approach, so I will just post my non-Maple calculation here.

@vv That's much better than my brute-force calculation, and also more correct in accounting for the order of tangency.  Vote up!

Your equation is y' = (1-x)*y^(-1).  That's the Benoulli equation with P(x)=0, Q(x)=1-x, n=-1.

@imparter 

There have been over 30 exchanges in this thread, and we are not yet close to obtaining the results that you are looking for. It seems to me that you are attempting to program in Maple at a level which is beyond your current state of knowledge. Perhaps you should set this project aside and focus on learning the basics of Maple programming, and then tackle this more advanced problem after gaining sufficient mastery of the basics.  Attempting to force your way through solving this particular problem is not a good way of learning Maple.

@imparter A few more errors fixed.  Pay closer attention to the details.

Query-Comments2.mw

@imparter In the attached worksheet I have pointed out a few issues that require your attention.  See if you can fixed those.

Query-Comments.mw

@imparter This worksheet handles the two sets of data through a for-loop.

Two_data-Fixed.mw

@imparter Upload your worksheet so that I may modify it.

@imparter I don't see a question in what you have written, so I can't tell what sort of response will be of help to you.  But for one thing, I see that you have not included the linestyle=dash in your code that I had suggested.  Did you try that?

@imparter The plot option linestyle=dash will plot with dashed lines.  Additionally, you may specify a list of colors for the graphs if you don't like the default colors.  So you may replace your current plot commands with

my_plot_1 := plot([%], r=0..h(z), color=["Red", "Green", "Blue", "Cyan"], linestyle=solid);
...
my_plot_2 := plot([%], r=0..h(z), color=["Red", "Green", "Blue", "Cyan"], linestyle=dash);

and then superimpose the two plots with

plots:-display(my_plot_1, my_plot_2);

Posting a picture of your worksheet is useless because I can look at it but cannot execute it. To get helpful responses, upload your Maple worksheet.  How?  In the message editor's toolbar, click on the big fat green arrow to upload your worksheet.

@imparter I don't know much about HPM and honestly, am not much interested in it.  From the page of calculations that you have supplied, it seems to me that the calculations are elementary and can be done easily by hand, if you know what you are doing, which I don't.  You don't need Maple for those calculations.

@imparter The purpose of your calculations is not clear to me.

1. You are looking for an approximate solution.  It seems to me that you are seeking an asymptotic expansion in terms of the parameter p.  That will be good for small values of p. Is that what you are after?
2. You say (r^2 - h^2)/4 is an approximate solution.  That doesn't look correct to me.  If you plug that expression into the differential equaiton, the result is 1/2.  It should be something like zero.  Do you have a typo there?
3. You have not specified boundary conditions.  Are you looking for a general solution?  You have defined an expression labeled ibvc which looks like an attempt to specify boundary conditions, but that doesn't do that.

@emendes In the file charsets/v2.0/charsets we have 100+ occurrences of the double-quote character (").  In modern Maple this has been replaced with the percent sign (%).  So the first thing to do is to replace all " with % in the file v2.0/charsets.  (Use your editor's global find/replace for that.)

There is one additional error to fix.  At the very end of v2.0/charsets we have

save `charsets.m`;

That's not good.  The syntax for Maple's save command is

save name1, name2, ..., namek, fileName

The variable names name1, etc., are missing.  I don't know what this application is supposed to do, so I cannot supply those variable names.  Maybe you can.  This may be as simple as uncommenting the commented-out save command near the end of v2.0/charsets.  Try it.  I haven't.

You have w(r,z) for the unknown of the differential equation.  However z appears neither in the differential equation nor in the boundary conditions.  Shouldn't that be just w(r)?