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DEtools

 DEplot_polygon
 generate the plot of the Newton polygon of a linear differential operator at a point

 Calling Sequence DEplot_polygon(L, y, (x = x0))

Parameters

 L - linear homogeneous differential equation y - unknown function to search for x0 - (optional) irreducible polynomial or infinity

Description

 • The DEplot_polygon function computes a plot of the Newton polygon of a linear differential operator at the point x0. The linear differential operator L corresponds to the differential equation $L\left(y\right)=0$.
 • The equation $L\left(y\right)=0$ must be homogeneous and linear in y and its derivatives, and its coefficients must be rational functions in the dependent variable x.
 • x0 must be a rational or an algebraic number or the symbol infinity. If x0 is not passed as an argument, x0 = 0 is assumed.
 • This function is part of the DEtools package, and so it can be used in the form DEplot_polygon(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[DEplot_polygon](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{ode}≔\left(\frac{{ⅆ}^{4}}{ⅆ{x}^{4}}y\left(x\right)\right){x}^{7}-\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)\left(x+{x}^{7}\right)-y\left(x\right){x}^{9}$
 ${\mathrm{ode}}{≔}\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}{}{y}{}\left({x}\right)\right){}{{x}}^{{7}}{-}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){}\left({{x}}^{{7}}{+}{x}\right){-}{y}{}\left({x}\right){}{{x}}^{{9}}$ (1)
 > $\mathrm{DEplot_polygon}\left(\mathrm{ode},y\left(x\right)\right)$

The command to create the plot from the Plotting Guide is

 > $\mathrm{DEplot_polygon}\left(\mathrm{ode},y\left(x\right),x=\mathrm{∞}\right)$