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](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\,x\right)x^7-\mathrm{diff}\left(y\left(x\right)\,x\right)\left(x+x^7\right)-y\left(x\right)x^9$

$\mathrm{ode}≔\left(\frac{{\ⅆ}^4}{\ⅆx^4}y\left(x\right)\right)x^7-\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(x^7+x\right)-y\left(x\right)x^9$

$\mathrm{DEplot\_polygon}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\mathrm{DEplot\_polygon}\left(\mathrm{ode}\,y\left(x\right)\,x=\mathrm{\infty }\right)$