**Q1**: In class we have discussed the problem of soap film
between to coaxial rings. The soap film takes a shape that minimizes the total
surface energy,

E = g Swhere g is the surface energy per unit area and S is the total area. The minimization is subject to the constraints that the the film spans a distance b and the ring radius is a.

(1) Implement a matlab or Mathematica function that computes the surface area of the film, with a given functional form of soap film by r(x), such that r(0)=r(b)=a. Give a test example to show that your program is correct.

(2) Verify that the solution

r(x) = c1 cosh((x-c2)/c1)satisfies the differential equation

r r'' - (r')^2 - 1 = 0

(3) This equation is the solution of minimum E or S. c2 is found to be b/2 (due to the requirement that r(0)=r(b)=a). Find the c1/a numerically by solving the equation

a = c1 cosh(b/(2c1))for each b/a, using some of the methods you learnt in numerical methods I or II. Use the solution data to draw the folding catastrophe bifurcation diagram (i.e., a diagram of b/a vs c1/a). c1 is the minimum radius at the center of film. You can measure all quantities in units of the ring radius a.

(4) Determine the stability of the fold catastrophe curve numerically (actually you need this result to draw the diagram in (3); solid line for stable equilibrium and dotted line for unstable equilibrium). To do this, compute the value S using the solution in (3) and program in (1), and compute the same with a slightly perturbed curve (by addition a small function dr(x), such that dr(0)=dr(b)=0). Compare the results to determine if it is a minimum or maximum.