|
<(K_r t^-r + ... + K1t^-1 + H + H1t + ... + H_m t^m) + [p](t)t^m+1,
and hence an identity
u = [psi](D + a)[K_r D^-r + ... + K1D^-1 + H + H1D + ... + H_m D^m]u + [p](D)D^m+1 u;
in this, since u contains no power of x higher than x^m, the second
term on the right may be omitted. We thus reach the conclusion that a
solution of the differential equation [psi](D + a)z = u is given by
z = (K_r D^-r + ... + K1D^-1 + H + H1D + ... + H_m D^m)u,
of which the operator on the right is obtained simply by expanding
1/[psi](D + a) in ascending powers of D, as if D were a numerical
quantity, the expansion being carried as far as the highest power of D
which, operating upon u, does not give zero. In this form every term
in z is capable of immediate calculation.
_Example._--For the equation
d^4v d^2y
---- + 2--- + y = x^3 cos x or (D^2 + 1)^2y = x^3 cos x,
dx^4 dx^3
the roots of the associated algebraic equation ([t]^2+1)^2 = 0 are [t] =
[+-]i, each repeated; the complementary function is thus
(A + Bx)e^ix + (C + Dx)e^ix,
where A, B, C, D are arbitrary constants; this is the same as
(H + Kx) cos x + (M + Nx) sin x,
where H, K, M, N are arbitrary constants. To obtain a particular
integral we must find a value of (1 + D^2)^-2 x^3 cos x; this is the
real part of (1 + D^2)^-2 e^ix x^3 and hence of e^ix [1 + (D + i)^2]^-2
x^3
or e^ix [2iD(1 + 1/2iD)]^-2 x^3,
or -1/4e^ix D^-2 (1 + iD - 3/4D^2 - 1/2iD^3 + 5/16 D^4 + 3/16 iD^5 ...)x^3,
or -1/4e^ix(1/20 x^5 + 1/4ix^4 - 3/4x^3 - 3/2 ix^2 + 15/8 x + 9/8 i);
the real part of this is
-1/4(1/20 x^5 - 3/4x^2 + 15/8 x) cos x + 1/4(1/4x^4 - 3/2 x^2 + 9/8) sin x.
This expression added to the complementary function found above gives
the complete integral; and no generality is lost by omitting from the
particular integral the terms -15/32 x cos x + 9/32 sin x, which are
of the types of terms already occurring in the complementary function.
The symbolical method which has been explained has wider applications
than that to which we have, for simplicity of explanation, restricted
it. For example, if [psi](x) be any function of x, and a1, a2, ... an
be different constants, and [(t + a1) (t + a2) ... (t + an)]^-1 when
expressed in partial fractions be written [Sigma]c_m(t + a_m)^-1, a
particular integra]]>
|