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104 2. Introductory applications

Finally, some examples of higher-order equations, which exhibit boundary-layer-type
solutions, are discussed in Smith (1985) and O’Malley (1991).

EXERCISES
Q2.1 Quadratic equations. Write down the exact roots of these quadratic equations,
where is a positive parameter.
(a) (b) (c)
Now, in each case, use the binomial expansion to obtain power-series repre-
sentations of these roots, valid for writing down the first three terms
for each root. (You may wish to investigate how these same expansions can be
derived directly from the original quadratic equations.)
Finally, obtain the corresponding power series which are valid for
Q2.2 Equations I. Find the first two terms in the asymptotic expansions of all the
real roots of these equations, for
(a) (b) (c)
(d) (e)
(f) (g)
(h) (i)
(j)
Q2.3 Equations II. Repeat Q2.2 for these slightly more involved equations.
(a)

(b) (c)

(d) (e)
Q2.4 Kepler’s equation. A routine problem in celestial mechanics is to find the eccentric
anomaly, u, given both the eccentricity and the mean anomaly nt
(where t is time measured from where u = 0, and where P is the
period); u is then the solution of Kepler’s equation

(see e.g. Boccaletti & Pucacco, 1996). For many orbits (for example, most
planets in our solar system), the eccentricity is very small; find the first three
terms in the asymptotic solution for u as Confirm that your 3-term
expansion is uniformly valid for all nt.
Q2.5 Complex roots. Find the first two terms in the asymptotic expansions of all the
roots of these equations, for
(a) (b) (c)

(d) (e) (f)