$SPAD/src/input kovacic.input

Timothy Daly December 3, 2016

Abstract

Contents

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1 License

— license —

--Copyright The Numerical Algorithms Group Limited 1991.

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— * —

)set break resume
)spool kovacic.output
)set message test on
)set message auto off
)clear all
--S 1 of 3
y := operator y
--R
--R
--R   (1)  y
--R                                                          Type: BasicOperator
--E 1

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I’ve added case 2 of Kovacic’s algorithm for version 1.5 (version 1.0 had case 1 only, version 2.0 will have case 3). For order 2 equations, we can now find those solutions whose logarithmic derivative is an algebraic function of degree 2 (version 1.0 found the solutions whose logarithmic derivatives were rational functions).

Here is an example of an equation that we can solve in 1.5 and not in 1.0: — * —

--S 2 of 3
eq := 2*x**3 * differentiate(y x,x,2) + 3*x**2 * differentiate(y x,x) - 2 * y x
--R
--R
--R          3 ,,        2 ,
--R   (2)  2x y  (x) + 3x y (x) - 2y(x)
--R
--R                                                    Type: Expression(Integer)
--E 2
--S 3 of 3
solve(eq,y,x).basis
--R
--R
--R               2      2
--R           - ----   ----
--R              +-+    +-+

--R             \|x    \|x
--R   (3)  [%e      ,%e    ]
--R                                              Type: List(Expression(Integer))
--E 3
)spool
)lisp (bye)

———-

References^{[1] nothing}

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