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.

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

$\mathrm{ode}≔y\left(x\right)-2x^2\mathrm{diff}\left(y\left(x\right)\,x\right)-6x^3\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-2x^4\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+x^6\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,4\right)\right)$

$\mathrm{ode}≔y\left(x\right)-2x^2\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-6x^3\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-2x^4\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+x^6\left(\frac{{\ⅆ}^4}{\ⅆx^4}y\left(x\right)\right)$

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

$\left[\left[\frac{1}{3}\,1-2u\right]\,\left[1\,-2+u\right]\right]$

$r≔\mathrm{regular\_parts}\left(\mathrm{ode}\,y\left(x\right)\,t\right)$

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

$\mathrm{ode1}≔\mathrm{op}\left(1\,\mathrm{select}\left(\mathrm{type}\,\mathrm{rhs}\left(r\left[1\right]\left[2\right]\right)\,\mathrm{DESol}\right)\right)\left[1\right]$

$\mathrm{ode1}≔\left(-\frac{20}{27}{t}^4+\frac{25}{9}{t}^3-\frac{46}{27}{t}^2+\frac{5}{36}t-\frac{20}{81}{t}^5\right)y\left(t\right)+\left(\frac{2}{27}{t}^6+\frac{26}{27}{t}^5-\frac{2}{3}{t}^2-\frac{4}{3}{t}^4-t+\frac{2}{27}{t}^3\right)\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)+\left(\frac{4}{81}{t}^7-\frac{1}{3}{t}^6+\frac{1}{6}{t}^5-\frac{1}{3}{t}^3-\frac{2}{9}{t}^4\right)\left(\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\right)+\left(-\frac{2}{81}{t}^8+\frac{1}{27}{t}^7-\frac{1}{27}{t}^5\right)\left(\frac{{\ⅆ}^3}{\ⅆt^3}y\left(t\right)\right)+\frac{t^9\left(\frac{{\ⅆ}^4}{\ⅆt^4}y\left(t\right)\right)}{324}$

$\mathrm{ode2}≔\mathrm{op}\left(1\,\mathrm{select}\left(\mathrm{type}\,\mathrm{rhs}\left(r\left[2\right]\left[2\right]\right)\,\mathrm{DESol}\right)\right)\left[1\right]$

$\mathrm{ode2}≔\left(3024t^3+1230t^2+141t\right)y\left(t\right)+\left(2016t^4+802t^3+120t^2+8t\right)\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)+\left(432t^5+132t^4+12t^3\right)\left(\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\right)+\left(36t^6+6t^5\right)\left(\frac{{\ⅆ}^3}{\ⅆt^3}y\left(t\right)\right)+t^7\left(\frac{{\ⅆ}^4}{\ⅆt^4}y\left(t\right)\right)$

$\mathrm{newton\_polygon}\left(\mathrm{ode1}\,y\left(t\right)\,u\right)$

$\left[\left[0\,-u\right]\,\left[1\,-u^2-9u-27\right]\,\left[3\,-12+u\right]\right]$

$\mathrm{newton\_polygon}\left(\mathrm{ode2}\,y\left(t\right)\,u\right)$

$\left[\left[0\,u\right]\,\left[1\,u^3+6u^2+12u+8\right]\right]$

$\mathrm{ode}≔\left(\frac{1}{x^{12}}-\frac{15}{x^9}+\frac{38}{x^6}-\frac{6}{x^3}\right)y\left(x\right)+\left(\frac{3}{x^8}-\frac{18}{x^5}+\frac{6}{x^2}\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+\left(\frac{3}{x^4}-\frac{3}{x}\right)\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x\right)+\mathrm{diff}\left(\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x\right)\,x\right)$

$\mathrm{ode}≔\left(\frac{1}{x^{12}}-\frac{15}{x^9}+\frac{38}{x^6}-\frac{6}{x^3}\right)y\left(x\right)+\left(\frac{3}{x^8}-\frac{18}{x^5}+\frac{6}{x^2}\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(\frac{3}{x^4}-\frac{3}{x}\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)$

$r≔\mathrm{regular\_parts}\left(\mathrm{ode}\,y\left(x\right)\,t\right)$

$r≔\left[\left[x\left(t\right)=t\,y\left(t\right)={\ⅇ}^{\frac{1}{3t^3}}t\mathrm{DESol}\left(\left\{t^3\left(\frac{{\ⅆ}^3}{\ⅆt^3}y\left(t\right)\right)\right\}\,\left\{y\left(t\right)\right\}\right)\right]\right]$

$\mathrm{simplify}\left(\mathrm{subs}\left(y\left(x\right)=\exp \left(\frac{1}{3x^3}\right)x\left(a+bx+c{x}^2\right)\,\mathrm{ode}\right)\right)$

$\frac{\left(\frac{{\partial }^3}{\partial x^3}\left({\ⅇ}^{\frac{1}{3x^3}}x\left(c{x}^2+bx+a\right)\right)\right)x^{11}+3\left(-x^{10}+x^7\right)\left(\frac{{\partial }^2}{\partial x^2}\left({\ⅇ}^{\frac{1}{3x^3}}x\left(c{x}^2+bx+a\right)\right)\right)+3\left(2x^9-6x^6+x^3\right)\left(\frac{\partial }{\partial x}\left({\ⅇ}^{\frac{1}{3x^3}}x\left(c{x}^2+bx+a\right)\right)\right)-6\left(c{x}^2+bx+a\right){\ⅇ}^{\frac{1}{3x^3}}\left(x^6-6x^3+\frac{1}{2}\right)\left(x^3-\frac{1}{3}\right)}{x^{11}}$