#### PolynomialSolutions

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\centerline{\fbox{\parbox{10.3cm}{\Large\bf UserFunction PolynomialSolutions} }}
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\vspace*{5mm}\noindent{\bf PolynomialSolutions}$(u:LODE(RATF Q,y,x))$
The single argument $u$ is a linear ordinary differential equation for the unknown function $y$. Its  coefficients are rational functions in the independent variable $x$. The meaning of the parameters is as follows.

$C$: Coefficient type.

$y$: Dependent variable.

$x$: Independent variable.

\vspace*{1mm}\noindent{\bf Specification.}
The maximal number of independent polynomial solutions
of degree not higher than 8 are returned.

\vspace*{1mm}\noindent{\bf Examples.}
The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window.

\4{\tt deq:=Df(y,x,3)-(4*x**2+2)/(x**3+x)*Df(y,x,2) }

\hspace*{10mm}
{\tt +(10*x**2+4)/(x**4+x**2)*Df(y,x)-(12*x**2+4)/(x**5+x**3)*y=0;}

\4{\tt ps:=PolynomialSolutions deq;}

\vspace{1mm}\hspace*{30mm}\fbox{\parbox{11cm}{

The Differential Equation:

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${y'''}-\fracsm{4x^{2}+2}{x^{3}+x}{y''}+\fracsm{10x^{2}+4}{x^{4}+x^{2}}{y'}-\fracsm{12x^{2}+4}{x^{5} +x^{3}}{y} = 0$

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Independent set of polynomial solutions

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$y=\{x^{2},\1x^{3}+x\}$  }}

\4{\tt deq:=Df(x,x,2)-1/x*Df(y,x)-144/x**2*y=0;}

\4{\tt PolynomialSolutions deq;}

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The Differential Equation:
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${y''}+\fracsm{1}{x}{y'}-\fracsm{144}{x^{2}}{y} = 0$
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Degree bound for polynomial solutions is 12.
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Equation does not have polynomial solutions of degree $\leq 9$  }}

\vspace*{5mm}\noindent{\bf PolynomialSolutions}$(u:LODE(RATF Q,y,x),n:BASIC)$
The first argument is the same as above. The second argument is a positive integer.

$n$: Positive integer.

\vspace*{1mm}\noindent{\bf Specification.}
The maximal number of independent polynomial solutions
of degree not higher than $n$ are returned.

\vspace*{1mm}\noindent{\bf Example.}
The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window.

\4{\tt deq:=Df(x,x,2)-1/x*Df(y,x)-144/x**2*y=0;}

\4{\tt PolynomialSolutions(deq,12);}

\vspace{1mm}\hspace*{30mm}\fbox{\parbox{11cm}{

The Differential Equation:
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${y''}+\fracsm{1}{x}{y'}-\fracsm{144}{x^{2}}{y} = 0$

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Independent set of polynomial solutions:
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$y=\{x^{12}\}$  }}

\vspace*{1mm}\noindent{\bf Related Functions:} RationalSolutions.

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{\bf References}

F.~Schwarz, {\em Algorithmic Lie Theory for Solving Ordinary Differential Equations}, CRC Press, 2007.

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