Mathematical Methods for Physicists: A concise introduction - Site Map

SECOND-ORDER EQUATIONS WITH CONSTANT COEFFICIENTS
We can form a polynomial function of D and write
F…D† ˆa 0 D n ‡ a 1 D n1 ‡‡a n1 D ‡ a n
so that
F…D† f …t† ˆa 0 D n f ‡ a 1 D n1 f ‡‡a n1 Df ‡ a n f
and we can interpret D 1 as follows
D 1 Df …t† ˆf …t†
and
Z
…Df †dt ˆ f :
Hence D 1 indicates the operation of integration (the inverse of di€erentiation).
Similarly D m f means `integrate f …t†m times'.
These properties of the linear operator D can be used to ®nd the particular
integral of Eq. (2.15):
d 2 y
dt 2 ‡ a dy
dt ‡ by ˆ
D2 ‡ aD ‡ b y ˆ f …t†
from which we obtain
where
y ˆ
1
D 2 ‡ aD ‡ b f …t† ˆ 1
F…D† f …t†;
F…D† ˆD 2 ‡ aD ‡ b:
…2:22†
The trouble with Eq. (2.22) is that it contains an expression involving Ds in the
denominator. It requires a fair amount of practice to use Eq. (2.22) to express y in
terms of conventional functions. For this, there are several rules to help us.
Given a power series of D
and since D n e t ˆ n e t , it follows that
Thus we have
Rules for D operators
G…D† ˆa 0 ‡ a 1 D ‡‡a n D n ‡
G…D†e t ˆ…a 0 ‡ a 1 D ‡‡a n D n ‡†e t ˆ G…†e t :
Rule (a): G…D†e t ˆ G…†e t provided G…† is convergent.
When G…D† is the expansion of 1=F…D† this rule gives
1
F…D† et ˆ 1
F…† et provided F…† 6ˆ0:
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