Tasks:

1. Plot the direction field on SLOPES using a window [-2,+2]x[-2,+5]. Make a hard copy of the said direction field. Look for a GLOBAL pattern in the direction field as well as LOCAL patterns in various regions of the solution plane:
   

   Do you see solution curves on the plane that serve like asymptotes (spine) or "attractors" for the other solution curves nearby ? Do you also see regions where each solution curve "goes" its own way without being attracted to other solution curves in the vicinity ? Do you see any lines that serve as vertical or horizontal asymptotes for ALL solution curves on the solution plane ? Mark your observations on the hard copy by hand, showing lines, curves, and regions with the mentioned behavior patterns.
   

2. Plot several solution curves on the direction field on SLOPES to test/investigate your observations in Part-1. Try to put as many integral curves as you see fit in various parts of the solution plane for an exhaustive analysis of the entire solution space without crashing the software. Make a new hard copy of the screen with all the integral curves on it. It may be a good idea to make two hard copies of the integral curves; one with the direction field in the background and one without.
   

   Mark the key features, mentioned in Part-1, of the solution family on the hard copy by hand.
   

3. Repeat Part-1 and Part-2 for a window [-1,+1]x[-1,+1], focusing on the region near (0,0). Keep looking for a "spine" that behaves like an attractor (asymptote).
   

4. The ODE has a SINGULARITY at x=0, that is, the ODE "blows up", since the coefficient (2x+1)/x becomes undefined, when x=0. How does this feature of the ODE display itself in the topology of the solution space ? How do the solution curves BEHAVE as they come near the region(s) where x=0 ? How do you explain this behavior through (a) the ODE itself, (b) the general solution ?
   

5. Now go to TWIDDLE and create a generic function f(x;c) representing the one-parameter family of solutions (the general solution) for the ODE. Using a window [-1,+2]x[-0.5,+0.5] draw a sufficient number of solution curves for various values (negative,zero,positive) of the parameter "c". Make a hard copy of the screen. Mark the c-value for each curve by hand on the hard copy. Comment on the curve for c=0. How does this curve (c=0) relate to your observations in Parts~1-3 above ? Please explain. Can you write the equation of this special curve ? Does this curve belong to the solution family of the DE ? Does the equation of this curve satisfy the ODE itself ? Does this curve fit into the direction field for the ODE ? Please comment on these and similar questions.
   

   The solution curves created on TWIDDLE using the closed-form expression for the solution family should naturally fit into the direction field for the ODE, as well as all the integral curves created on SLOPES in Part-1, Part-2 and Part-3. Please comment on this anticipated correspondence.
   

6. Address the questions raised in Part-4 using the graphs made on TWIDDLE . Do your observations match your findings in Part-4 ? Please comment.
   

7. Repeat the analysis in Part-1 by zooming in on a much smaller window [-0.01,+0.01]x[-0.01,+0.01] to see if the global characteristics of the direction field are preserved at the local level around (0,0). Follow the same procedure as in Part-1.
   

8. What can you conclude about the impact of the singularity at x=0 in shaping the geometry of the solution curves in the vicinity of the singularity ? How does the spine at c=0 fit into this general picture ? Please explain.
   

   You may find it useful to zoom out and look at the direction field over a window [-2,+2]x[-4,+12] to see the BIG PICTURE away from (0,0) to answer this part.
   

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