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214 5. Some worked examples arising from physical problems
Figure 12. Circuit diagram for the diode oscillator with a current pump.
We seek the first term of an asymptotic expansion for
for fixed (and we may take advantage of small to simplify some of the details, but
this is not essential). Thus the problem for from (5.47), becomes
which can be solved exactly (by writing it is convenient to
express the initial value as
and then we obtain
The nature of this solution is, perhaps, not immediately apparent, but it becomes more
transparent if we invoke We provide approximations to for various t,
below (the details of which are left as an exercise):
