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The Slode Package

The Slode package contains functions to find formal power series solutions of linear ordinary differential equations (LODEs), to determine points for some special series solutions (hypergeometric, rational, polynomial, and sparse series).

$restart$

>	$with (Slode) &colon;$

Create a LODEstruct Structure from a LODE

DEdetermine - Check a LODE and create the LODEstruct structure

>	$ode := (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) (x - 1) - y (x) = 0$

$ode := (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) (x - 1) - y (x) = 0$

(1.1)

>	$DEdetermine (ode, y (x))$

$LODEstruct (\{(\frac{&DifferentialD;}{&DifferentialD; x} y (x)) (x - 1) - y (x) = 0\}, \{y (x)\})$

(1.2)

Create Formal Power/Taylor Series Solutions

FPseries - Create formal power series solutions

>	$ode := (3 x^{2} - 6 x + 3) (\frac{&DifferentialD;}{&DifferentialD; x} (\frac{&DifferentialD;}{&DifferentialD; x} y (x))) + (12 x - 12) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + 6 y (x)$

$ode := (3 x^{2} - 6 x + 3) (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) + (12 x - 12) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + 6 y (x)$

(2.1)

>	$FPseries (ode, y (x), v (n))$

$FPSstruct ({_C}_{0} + {_C}_{1} x + \sum_{n = 2}^{\infty} v (n) x^{n}, (n^{2} - n) v (n) + (- 2 n^{2} + 2 n) v (n - 1) + (n^{2} - n) v (n - 2))$

(2.2)

FTseries - Create formal Taylor series solutions

>	$FTseries (ode, y (x), v (n), 0,_A, 2)$

$FPSstruct ({_A}_{0} + {_A}_{1} x + \frac{1}{2} (- 2 {_A}_{0} + 4 {_A}_{1}) x^{2} + \sum_{n = 3}^{\infty} \frac{v (n) x^{n}}{n &excl;}, v (n) - 2 n v (n - 1) + (n^{2} - n) v (n - 2))$

(2.3)

Determine Candidate Points for Some Special Series Solutions

candidate_points - Determine candidate points for hypergeometric, rational and polynomial power series solutions of homogeneous LODEs with polynomial coefficients

>	$ode := (3 x^{2} - 6 x + 3) (\frac{&DifferentialD;}{&DifferentialD; x} (\frac{&DifferentialD;}{&DifferentialD; x} y (x))) + (12 x - 12) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + 6 y (x)$

$ode := (3 x^{2} - 6 x + 3) (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) + (12 x - 12) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + 6 y (x)$

(3.1)

>	$candidate_points (ode, y (x),' polynomial')$

$\{0\}$

(3.2)

>	$candidate_points (ode, y (x),' rational')$

$\{1\}$

(3.3)

>	$candidate_points (ode, y (x),' hypergeom')$

$\{1, any_ordinary_point\}$

(3.4)

candidate_mpoints - Determine mpoints for m-sparse power series

>	$ode := (2 + x^{2}) (\frac{{&DifferentialD;}^{3}}{&DifferentialD; x^{3}} y (x)) - 2 (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) x + (2 + x^{2}) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) - 2 x y (x)$

$ode := (2 + x^{2}) (\frac{{&DifferentialD;}^{3}}{&DifferentialD; x^{3}} y (x)) - 2 (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) x + (2 + x^{2}) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) - 2 x y (x)$

(3.5)

>	$candidate_mpoints (ode, y (x))$

$[[2, LODEstruct (\{y (x) + \frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)\}, \{y (x)\}), \{0\}]]$

(3.6)

candidate_mpoints/irreducible - Use a much faster algorithm that returns all m-points if the equation is irreducible

>	$ode := \frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x) + (x - 1) y (x)$

$ode := \frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x) + (x - 1) y (x)$

(3.7)

>	$candidate_mpoints (ode, y (x),' irreducible')$

$[[3, LODEstruct (\{y (x)\}, \{y (x)\}), \{1\}]]$

(3.8)

Find Formal Power Series Solutions

polynomial_series_sol - Find formal power series solutions with polynomial coefficients

>	$ode := (3 x^{2} - 6 x + 3) (\frac{&DifferentialD;}{&DifferentialD; x} (\frac{&DifferentialD;}{&DifferentialD; x} y (x))) + (12 x - 12) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + 6 y (x)$

$ode := (3 x^{2} - 6 x + 3) (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) + (12 x - 12) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + 6 y (x)$

(4.1)

>	$polynomial_series_sol (ode, y (x))$

$\{\sum_{n = 0}^{\infty} ({_C}_{0} + n {_C}_{1}) x^{n}\}$

(4.2)

rational_series_sol - Find formal power series solutions with rational coefficients

>	$ode := (3 - x) (\frac{&DifferentialD;}{&DifferentialD; x} (\frac{&DifferentialD;}{&DifferentialD; x} y (x))) - (\frac{&DifferentialD;}{&DifferentialD; x} y (x))$

$ode := (3 - x) (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) - (\frac{&DifferentialD;}{&DifferentialD; x} y (x))$

(4.3)

>	$rational_series_sol (ode, y (x))$

$\{{_C}_{1} + {_C}_{0} (\sum_{n = 1}^{\infty} \frac{{(x - 2)}^{n}}{n})\}$

(4.4)

hypergeom_series_sol - Find formal power series solutions with hypergeometric coefficients

>	$ode := 2 x (x - 1) (\frac{&DifferentialD;}{&DifferentialD; x} (\frac{&DifferentialD;}{&DifferentialD; x} y (x))) + (7 x - 3) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + 2 y (x) = 0$

$ode := 2 x (x - 1) (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) + (7 x - 3) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + 2 y (x) = 0$

(4.5)

>	$hypergeom_series_sol (ode, y (x))$

$\{{_C}_{1} (\sum_{n = 0}^{\infty} \frac{(n + 1) x^{n}}{2 n + 1}), {_C}_{1} (\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} Γ (n + \frac{1}{2}) {(x - 1)}^{n}}{Γ (n + 1)}), {_C}_{1} (\sum_{n = 0}^{\infty} \frac{Γ (n + \frac{1}{2}) {(x + 1)}^{n}}{n &excl;})\}$

(4.6)

msparse_series_sol - Find formal m-sparse power series solutions

$ode := \frac{2 (x^{3} - \frac{2 x^{2}}{3} + \frac{1 x}{9}) (\frac{{&DifferentialD;}^{3}}{&DifferentialD; x^{3}} y (x))}{9} + \frac{2 (9 x^{2} - 4 x + \frac{1}{3}) (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x))}{9} + \frac{2 (18 x - 4) (\frac{&DifferentialD;}{&DifferentialD; x} y (x))}{9} + \frac{4 y (x)}{3}$

$ode := \frac{1}{9} (2 x^{3} - \frac{4}{3} x^{2} + \frac{2}{9} x) (\frac{{&DifferentialD;}^{3}}{&DifferentialD; x^{3}} y (x)) + \frac{1}{9} (18 x^{2} - 8 x + \frac{2}{3}) (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) + \frac{1}{9} (36 x - 8) (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + \frac{4}{3} y (x)$

(4.7)

>	$msparse_series_sol (ode, y (x), v (n))$

$\{FPSstruct (\frac{1}{36} {_C}_{2} + {_C}_{2} {(x - \frac{1}{6})}^{2} + 36 {_C}_{2} {(x - \frac{1}{6})}^{4} + \sum_{n = 3}^{\infty} v (2 n) {(x - \frac{1}{6})}^{2 n}, - 36 v (2 n - 2) + v (2 n)), FPSstruct ({_C}_{1} (x - \frac{1}{6}) + 36 {_C}_{1} {(x - \frac{1}{6})}^{3} + 1296 {_C}_{1} {(x - \frac{1}{6})}^{5} + \sum_{n = 3}^{\infty} v (2 n + 1) {(x - \frac{1}{6})}^{2 n + 1}, - 36 v (2 n - 1) + v (2 n + 1))\}$

(4.8)

mhypergeom_series_sol - Find formal m-sparse m-hypergeometric power series

>	$ode := \frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x) + (x - 1) y (x)$

$ode := \frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x) + (x - 1) y (x)$

(4.9)

>	$mhypergeom_series_sol (ode, y (x))$

$\{{_C}_{1} (\sum_{n = 0}^{\infty} \frac{{(- \frac{1}{9})}^{n} {(x - 1)}^{3 n + 1}}{Γ (n + 1) Γ (n + \frac{4}{3})}), {_C}_{1} (\sum_{n = 0}^{\infty} \frac{{(- \frac{1}{9})}^{n} {(x - 1)}^{3 n}}{Γ (n + \frac{2}{3}) Γ (n + 1)})\}$

(4.10)

hypergeom_formal_sol - Find formal solutions with hypergeometric series

>	$ode := 3 x y (x) + (4 + 4 x) x (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + (- 3 - 3 x) x^{2} (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x))$

$ode := 3 x y (x) + (4 + 4 x) x (\frac{&DifferentialD;}{&DifferentialD; x} y (x)) + (- 3 - 3 x) x^{2} (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x))$

(4.11)

>	$hypergeom_formal_sol (ode, y (x), t, 0)$

$[[x (t) = t, y (t) = \frac{9}{4} \frac{Γ (\frac{2}{3}) (\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n + 1} Γ (n - \frac{7}{6} + \frac{1}{6} \sqrt{85}) Γ (n - \frac{7}{6} - \frac{1}{6} \sqrt{85}) t^{n}}{Γ (n - \frac{4}{3}) Γ (n + 1)})}{Γ (- \frac{1}{6} + \frac{1}{6} \sqrt{85}) Γ (- \frac{1}{6} - \frac{1}{6} \sqrt{85})}], [x (t) = t, y (t) = \frac{56}{81} \frac{t^{7 / 3} \sqrt{3} π (\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n + 1} Γ (n + \frac{7}{6} + \frac{1}{6} \sqrt{85}) Γ (n + \frac{7}{6} - \frac{1}{6} \sqrt{85}) t^{n}}{Γ (n + \frac{10}{3}) Γ (n + 1)})}{Γ (\frac{13}{6} + \frac{1}{6} \sqrt{85}) Γ (\frac{13}{6} - \frac{1}{6} \sqrt{85}) Γ (\frac{2}{3})}]]$

(4.12)

References

Abramov, S. "m-Sparse Solutions of Linear Ordinary Differential Equations with Polynomial Coefficients." Discrete Math., Vol. 217, (2000): 3-15.

Abramov, S. "Power Series Solutions with "Eventually Nice" Coefficients of Linear Ordinary Differential Equations." Proc. FPSAC'98. 1998.

Abramov, S. "Solutions of linear differential equations in the class of sparse power series." Proc. FPSAC'97, pp. 1-10. 1997.

Abramov, S.; Bronstein, M.; and Petkovsek, M. "On Polynomial Solutions of Linear Operator Equations." Proceedings of ISSAC '95. 1995.

Abramov, S., and Petkovsek, M. "Special Power Series Solutions Of Linear Differential Equations." Proceedings of FPSAC'96. 1996

Petkovsek, M., and Salvy, B. "Finding all hypergeometric solutions of linear differential equations." Proceedings of ISSAC '93. 1993

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