diffalg[power_series_solution] - expand the non-singular zero of a characterizable differential ideal into integral power series
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Calling Sequence
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power_series_solution (point, order, J, 'syst', 'params')
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Parameters
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point
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list or set of names or equations
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order
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non-negative integer
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J
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characterizable differential ideal
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syst
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(optional) name
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params
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(optional) name
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Description
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The function power_series_solution computes a formal integral power series solution of the differential system equations , inequations . Such a system is formally integrable. See the last example below.
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If point is a singular point of equations (J), then power_series_solution returns FAIL. Nevertheless, this does not mean that no formal power series solution exists at that point.
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When point is not singular, the series is truncated at the order given by the parameter order. They could be expanded up to any order, though convergence is not guaranteed.
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The series involve parameters corresponding to initial conditions to be given.
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The parameters appear as u, where u is a differential indeterminate if it represents the value of the solution at point, or _Cu_x, where x is some derivation variable, if it represents the value of the value of the first derivative of according to x at point.
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The parameters must satisfy a triangular system of polynomial equations and inequations given by syst in terms of the parameters involved in the power series solution.
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If present, the variable params receives the subset of the parameters involved in the power series solution that can almost be chosen arbitrarily if not for some inequations in syst.
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If J is a radical differential ideal represented by a list of characterizable differential ideals, the function power_series_solution is mapped on its component.
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The command with(diffalg,power_series_solution) allows the use of the abbreviated form of this command.
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Examples
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Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.
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Let us explain now why, in general, we have to start from a characterizable differential system instead of any differential system. Consider the differential system given by these two differential polynomials.
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We are looking for a solution starting as:
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It seems that we can choose an initial condition () and that, by differentiating the equations, all the coefficients in the expansion can be expressed in terms of .
The first terms do not lead to any problem:
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To compute the next term we can either differentiate or . The problem is that the results obtained are not compatible.
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The system is not formally integrable as it stands. The only solution of the system is:
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