LinearOrdinaryDifferentialOperator(A, diff) is the domain of linear ordinary differential operators with coefficients in a ring A with a given derivation.
Problem:
Find the first few coefficients of exp(x)/x^i of Dop phi where
Dop := D^3 + G/x^2 * D + H/x^3 - 1
phi := sum(s[i]*exp(x)/x^i, i = 0..)
Solution:
Define the differential.
Dx: LODO(EXPR INT, f +-> D(f, x))
Type: Void
Dx := D()
D
Type: LinearOrdinaryDifferentialOperator(Expression Integer,
theMap LAMBDA-CLOSURE(NIL,NIL,NIL,G1404 envArg,
SPADCALL(G1404,QUOTE x,
ELT(*1;anonymousFunction;0;frame0;internal;MV,0))))
Now define the differential operator Dop.
Dop:= Dx^3 + G/x^2*Dx + H/x^3 - 1
3
3 G - x + H
D + -- D + --------
2 3
x x
Type: LinearOrdinaryDifferentialOperator(Expression Integer,
theMap LAMBDA-CLOSURE(NIL,NIL,NIL,G1404 envArg,
SPADCALL(G1404,QUOTE x,
ELT(*1;anonymousFunction;0;frame0;internal;MV,0))))
n == 3
Type: Void
phi == reduce(+,[subscript(s,[i])*exp(x)/x^i for i in 0..n])
Type: Void
phi1 == Dop(phi) / exp x
Type: Void
phi2 == phi1 *x**(n+3)
Type: Void
phi3 == retract(phi2)@(POLY INT)
Type: Void
pans == phi3 ::UP(x,POLY INT)
Type: Void
pans1 == [coefficient(pans, (n+3-i) :: NNI) for i in 2..n+1]
Type: Void
leq == solve(pans1,[subscript(s,[i]) for i in 1..n])
Type: Void
Evaluate this for several values of n.
leq
2 3 2
s G 3s H + s G + 6s G (9s G + 54s )H + s G + 18s G + 72s G
0 0 0 0 0 0 0 0 0
[[s = ---,s = ------------------,s = --------------------------------------]]
1 3 2 18 3 162
Type: List List Equation Fraction Polynomial Integer
n==4
Type: Void
leq
[
2
s G 3s H + s G + 6s G
0 0 0 0
[s = ---, s = ------------------,
1 3 2 18
3 2
(9s G + 54s )H + s G + 18s G + 72s G
0 0 0 0 0
s = --------------------------------------,
3 162
s =
4
2 2 4 3 2
27s H + (18s G + 378s G + 1296s )H + s G + 36s G + 396s G
0 0 0 0 0 0 0
+
1296s G
0
/
1944
]
]
Type: List List Equation Fraction Polynomial Integer
n==7
Type: Void
leq
[
2
s G 3s H + s G + 6s G
0 0 0 0
[s = ---, s = ------------------,
1 3 2 18
3 2
(9s G + 54s )H + s G + 18s G + 72s G
0 0 0 0 0
s = --------------------------------------,
3 162
s =
4
2 2 4 3 2
27s H + (18s G + 378s G + 1296s )H + s G + 36s G + 396s G
0 0 0 0 0 0 0
+
1296s G
0
/
1944
,
s =
5
2 3 2
(135s G + 2268s )H + (30s G + 1350s G + 16416s G + 38880s )H
0 0 0 0 0 0
+
5 4 3 2
s G + 60s G + 1188s G + 9504s G + 25920s G
0 0 0 0 0
/
29160
,
s =
6
3 2 2
405s H + (405s G + 18468s G + 174960s )H
0 0 0 0
+
4 3 2 6
(45s G + 3510s G + 88776s G + 777600s G + 1166400s )H + s G
0 0 0 0 0 0
+
5 4 3 2
90s G + 2628s G + 27864s G + 90720s G
0 0 0 0
/
524880
,
s =
7
3
(2835s G + 91854s )H
0 0
+
3 2 2
(945s G + 81648s G + 2082996s G + 14171760s )H
0 0 0 0
+
5 4 3 2
(63s G + 7560s G + 317520s G + 5554008s G + 34058880s G)H
0 0 0 0 0
+
7 6 5 4 3 2
s G + 126s G + 4788s G + 25272s G - 1744416s G - 26827200s G
0 0 0 0 0 0
+
- 97977600s G
0
/
11022480
]
]
Type: List List Equation Fraction Polynomial Integer
See Also: