Rit (Vísindafélag Íslendinga) - 01.06.1946, Blaðsíða 22

22
3. MATTHIEU’S DIFFERENTIAL EQUATION
I have above briefly set forth the main points of the theory
underlying the method of asymptotic integration of differen-
tial equations. I shall now apply the method to a few
examples in order to show how the formulas given above
actually make a rather easy insight into the real meaning
of the differential equations even when they are of such
a form as to be difficult to deal with by the ordinary me-
thods. Certainly the solution requires a considerable work
of calculation, but the method is straightforward and does
not call for mathematical tricks, although we may in special
cases make the work easier by applying a suitable proce-
dure.
As a first example I choose Matthieu’s differential equa-
tion which I write
(36)
For the method of asymptotic integration it is more advan-
tageous to add the phase t to t but we could as well begin
with the common form of Matthieu’s equation where t = 0.
It is advisable to begin with applying the asymptotic
differentiation as given in (20). Then we get from (36)
after an obvious transformation
(37)
or
k (k — 1) -)- ot (l-j-f Cos(t-(-T))
(37a)