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 regular_parts
 Find regular parts of a linear ode

 Calling Sequence regular_parts(L, y, t, [x=x0])

Parameters

 L - linear homogeneous differential equation y - unknown function to search for t - name used as parametrization variable x0 - (optional) a rational, an algebraic number or infinity

Description

 • The regular_parts function computes the minimal generalized exponents of L at the point x0 and the corresponding regular parts. These are operators L_e which result from L by replacing y(x) by exp(int(e, x))*y(x). The Newton polygon of L_e at x_0 has a segment of slope 0 and 0 is a root of the indicial polynomial.
 • 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 variable x.
 • x0 must be a rational or an algebraic number or the symbol infinity. If x0 is not passed as argument, x0 = 0 is assumed.
 • The output is a set of solutions which are of the form exp(int(e, x))*y where e is a minimal generalized exponent and y is given as DESol object.
 • The command with(DEtools,regular_parts) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{ode}≔y\left(x\right)-2{x}^{2}\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)-6{x}^{3}\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)-2{x}^{4}\left(\frac{{ⅆ}^{3}}{ⅆ{x}^{3}}y\left(x\right)\right)+{x}^{6}\left(\frac{{ⅆ}^{4}}{ⅆ{x}^{4}}y\left(x\right)\right)$
 ${\mathrm{ode}}{≔}{y}{}\left({x}\right){-}{2}{}{{x}}^{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{6}{}{{x}}^{{3}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{2}{}{{x}}^{{4}}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{{x}}^{{6}}{}\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (1)

Then 0 is a singular point of this equation. Newton polygon is:

 > $\mathrm{newton_polygon}\left(\mathrm{ode},y\left(x\right),u\right)$
 $\left[\left[\frac{{1}}{{3}}{,}{1}{-}{2}{}{u}\right]{,}\left[{1}{,}{-}{2}{+}{u}\right]\right]$ (2)

There are slopes > 0 so 0 is an irregular singular point.

 > $r≔\mathrm{regular_parts}\left(\mathrm{ode},y\left(x\right),t\right)$
 ${r}{≔}\left[\left[{x}{}\left({t}\right){=}\frac{{{t}}^{{3}}}{{2}}{,}{y}{}\left({t}\right){=}{{ⅇ}}^{{-}\frac{{3}}{{t}}}{}{t}{}{\mathrm{DESol}}{}\left(\left\{\left({-}\frac{{20}}{{27}}{}{{t}}^{{4}}{+}\frac{{25}}{{9}}{}{{t}}^{{3}}{-}\frac{{46}}{{27}}{}{{t}}^{{2}}{+}\frac{{5}}{{36}}{}{t}{-}\frac{{20}}{{81}}{}{{t}}^{{5}}\right){}{y}{}\left({t}\right){+}\left(\frac{{2}}{{27}}{}{{t}}^{{6}}{+}\frac{{26}}{{27}}{}{{t}}^{{5}}{-}\frac{{2}}{{3}}{}{{t}}^{{2}}{-}\frac{{4}}{{3}}{}{{t}}^{{4}}{-}{t}{+}\frac{{2}}{{27}}{}{{t}}^{{3}}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}\left(\frac{{4}}{{81}}{}{{t}}^{{7}}{-}\frac{{1}}{{3}}{}{{t}}^{{6}}{+}\frac{{1}}{{6}}{}{{t}}^{{5}}{-}\frac{{1}}{{3}}{}{{t}}^{{3}}{-}\frac{{2}}{{9}}{}{{t}}^{{4}}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}\left({-}\frac{{2}}{{81}}{}{{t}}^{{8}}{+}\frac{{1}}{{27}}{}{{t}}^{{7}}{-}\frac{{1}}{{27}}{}{{t}}^{{5}}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{t}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}\frac{{{t}}^{{9}}{}\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{t}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}{{324}}\right\}{,}\left\{{y}{}\left({t}\right)\right\}\right)\right]{,}\left[{x}{}\left({t}\right){=}{t}{,}{y}{}\left({t}\right){=}{{ⅇ}}^{{-}\frac{{2}}{{t}}}{}{{t}}^{{9}}{}{\mathrm{DESol}}{}\left(\left\{\left({3024}{}{{t}}^{{3}}{+}{1230}{}{{t}}^{{2}}{+}{141}{}{t}\right){}{y}{}\left({t}\right){+}\left({2016}{}{{t}}^{{4}}{+}{802}{}{{t}}^{{3}}{+}{120}{}{{t}}^{{2}}{+}{8}{}{t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}\left({432}{}{{t}}^{{5}}{+}{132}{}{{t}}^{{4}}{+}{12}{}{{t}}^{{3}}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}\left({36}{}{{t}}^{{6}}{+}{6}{}{{t}}^{{5}}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{t}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}{{t}}^{{7}}{}\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{t}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)\right\}{,}\left\{{y}{}\left({t}\right)\right\}\right)\right]\right]$ (3)

yields two transformed differential equations:

 > $\mathrm{ode1}≔{\mathrm{op}\left(1,\mathrm{select}\left(\mathrm{type},\mathrm{rhs}\left({{r}_{1}}_{2}\right),\mathrm{DESol}\right)\right)}_{1}$
 ${\mathrm{ode1}}{≔}\left({-}\frac{{20}}{{27}}{}{{t}}^{{4}}{+}\frac{{25}}{{9}}{}{{t}}^{{3}}{-}\frac{{46}}{{27}}{}{{t}}^{{2}}{+}\frac{{5}}{{36}}{}{t}{-}\frac{{20}}{{81}}{}{{t}}^{{5}}\right){}{y}{}\left({t}\right){+}\left(\frac{{2}}{{27}}{}{{t}}^{{6}}{+}\frac{{26}}{{27}}{}{{t}}^{{5}}{-}\frac{{2}}{{3}}{}{{t}}^{{2}}{-}\frac{{4}}{{3}}{}{{t}}^{{4}}{-}{t}{+}\frac{{2}}{{27}}{}{{t}}^{{3}}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}\left(\frac{{4}}{{81}}{}{{t}}^{{7}}{-}\frac{{1}}{{3}}{}{{t}}^{{6}}{+}\frac{{1}}{{6}}{}{{t}}^{{5}}{-}\frac{{1}}{{3}}{}{{t}}^{{3}}{-}\frac{{2}}{{9}}{}{{t}}^{{4}}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}\left({-}\frac{{2}}{{81}}{}{{t}}^{{8}}{+}\frac{{1}}{{27}}{}{{t}}^{{7}}{-}\frac{{1}}{{27}}{}{{t}}^{{5}}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{t}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}\frac{{{t}}^{{9}}{}\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{t}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}{{324}}$ (4)
 > $\mathrm{ode2}≔{\mathrm{op}\left(1,\mathrm{select}\left(\mathrm{type},\mathrm{rhs}\left({{r}_{2}}_{2}\right),\mathrm{DESol}\right)\right)}_{1}$
 ${\mathrm{ode2}}{≔}\left({3024}{}{{t}}^{{3}}{+}{1230}{}{{t}}^{{2}}{+}{141}{}{t}\right){}{y}{}\left({t}\right){+}\left({2016}{}{{t}}^{{4}}{+}{802}{}{{t}}^{{3}}{+}{120}{}{{t}}^{{2}}{+}{8}{}{t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}\left({432}{}{{t}}^{{5}}{+}{132}{}{{t}}^{{4}}{+}{12}{}{{t}}^{{3}}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}\left({36}{}{{t}}^{{6}}{+}{6}{}{{t}}^{{5}}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{t}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}{{t}}^{{7}}{}\left(\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{t}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)$ (5)

These operators have a Newton polygon with slope 0:

 > $\mathrm{newton_polygon}\left(\mathrm{ode1},y\left(t\right),u\right)$
 $\left[\left[{0}{,}{-}{u}\right]{,}\left[{1}{,}{-}{{u}}^{{2}}{-}{9}{}{u}{-}{27}\right]{,}\left[{3}{,}{-}{12}{+}{u}\right]\right]$ (6)
 > $\mathrm{newton_polygon}\left(\mathrm{ode2},y\left(t\right),u\right)$
 $\left[\left[{0}{,}{u}\right]{,}\left[{1}{,}{{u}}^{{3}}{+}{6}{}{{u}}^{{2}}{+}{12}{}{u}{+}{8}\right]\right]$ (7)

This can help to find closed-form solutions:

 > $\mathrm{ode}≔\left(\frac{1}{{x}^{12}}-\frac{15}{{x}^{9}}+\frac{38}{{x}^{6}}-\frac{6}{{x}^{3}}\right)y\left(x\right)+\left(\frac{3}{{x}^{8}}-\frac{18}{{x}^{5}}+\frac{6}{{x}^{2}}\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+\left(\frac{3}{{x}^{4}}-\frac{3}{x}\right)\left(\frac{ⅆ}{ⅆx}\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)\right)+\frac{ⅆ}{ⅆx}\left(\frac{ⅆ}{ⅆx}\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)\right)$
 ${\mathrm{ode}}{≔}\left(\frac{{1}}{{{x}}^{{12}}}{-}\frac{{15}}{{{x}}^{{9}}}{+}\frac{{38}}{{{x}}^{{6}}}{-}\frac{{6}}{{{x}}^{{3}}}\right){}{y}{}\left({x}\right){+}\left(\frac{{3}}{{{x}}^{{8}}}{-}\frac{{18}}{{{x}}^{{5}}}{+}\frac{{6}}{{{x}}^{{2}}}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left(\frac{{3}}{{{x}}^{{4}}}{-}\frac{{3}}{{x}}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)$ (8)
 > $r≔\mathrm{regular_parts}\left(\mathrm{ode},y\left(x\right),t\right)$
 ${r}{≔}\left[\left[{x}{}\left({t}\right){=}{t}{,}{y}{}\left({t}\right){=}{{ⅇ}}^{\frac{{1}}{{3}{}{{t}}^{{3}}}}{}{t}{}{\mathrm{DESol}}{}\left(\left\{{{t}}^{{3}}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{t}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)\right\}{,}\left\{{y}{}\left({t}\right)\right\}\right)\right]\right]$ (9)

Since the general solution of the regular part is a+b*x+c*x^2 for some constants a,b and c, we obtain the general solution of the original equation by taking into account the exponential transformation:

 > $\mathrm{simplify}\left(\mathrm{subs}\left(y\left(x\right)={ⅇ}^{\frac{1}{3{x}^{3}}}x\left(a+bx+c{x}^{2}\right),\mathrm{ode}\right)\right)$
 $\frac{\left(\frac{{{\partial }}^{{3}}}{{\partial }{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({{ⅇ}}^{\frac{{1}}{{3}{}{{x}}^{{3}}}}{}{x}{}\left({c}{}{{x}}^{{2}}{+}{b}{}{x}{+}{a}\right)\right)\right){}{{x}}^{{11}}{+}\left({-}{3}{}{{x}}^{{10}}{+}{3}{}{{x}}^{{7}}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({{ⅇ}}^{\frac{{1}}{{3}{}{{x}}^{{3}}}}{}{x}{}\left({c}{}{{x}}^{{2}}{+}{b}{}{x}{+}{a}\right)\right)\right){+}\left({6}{}{{x}}^{{9}}{-}{18}{}{{x}}^{{6}}{+}{3}{}{{x}}^{{3}}\right){}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({{ⅇ}}^{\frac{{1}}{{3}{}{{x}}^{{3}}}}{}{x}{}\left({c}{}{{x}}^{{2}}{+}{b}{}{x}{+}{a}\right)\right)\right){-}{6}{}\left({{x}}^{{3}}{-}\frac{{1}}{{3}}\right){}\left({{x}}^{{6}}{-}{6}{}{{x}}^{{3}}{+}\frac{{1}}{{2}}\right){}\left({c}{}{{x}}^{{2}}{+}{b}{}{x}{+}{a}\right){}{{ⅇ}}^{\frac{{1}}{{3}{}{{x}}^{{3}}}}}{{{x}}^{{11}}}$ (10)