The two highest coefficients of a symmetric power of a second order
operator.
Let L = D^2 + a(x) * D + b(x) where D = d/dx.
Then the n-1'st symmetric power of L is of the form
D^n + A.n * D^(n-1) + B.n * D^(n-2) + ....
In the Maple document below we prove a formula for A.n and B.n in terms of
a(x) and b(x), which one can invert to find a formula for a(x) and b(x)
in terms of A.n and B.n.
The latter formula can be used to detect very easily if an operator
L.n = D^n + A.n * D^(n-1) + B.n * D^(n-2) + ....
is a symmetric power of a second order operator L or not, and if so,
to find L.
This is useful for solving L.n: If L.n is a symmetric power of a
second order operator L, then solving L.n has been reduced to solving L.
If L.n is not a symmetric power of a second order operator L, then very
little CPU time will be lost because this test is so simple.
This algorithm is part of Maple and was implemented by George Labahn.
These formulas can be computed with the following
Maple document but were already known
earlier, see:
R. Chalkley,
Relative invariants for homogeneous linear differential equations,
J. Differential Equations 80, 107-153, 1989.