The ExtendRegularSolution(sol, deg) function returns the regular solution with initial terms of the Laurent series solution involved in sol extended to the specified formal degree deg.

The specified solution sol must be in the form returned by LinearFunctionalSystems[RegularSolution] or ExtendRegularSolution. In other words, sol must have an attribute of the special form as described in LinearFunctionalSystems[RegularSolution].

This function computes the additional terms of the involved series expansions using the invertible leading matrix of the matrix recurrence system corresponding to the linear functional system that was originally specified. These recurrences are part of the special structure stored in the attribute of the given solution.

The result of ExtendRegularSolution is returned in the same form as the result of LinearFunctionalSystems[RegularSolution].

This function is part of the LinearFunctionalSystems package, and so it can be used in the form ExtendRegularSolution(..) only after executing the command with(LinearFunctionalSystems). However, it can always be accessed through the long form of the command by using the form LinearFunctionalSystems[ExtendRegularSolution](..).

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

$\mathrm{sys}≔\left[x^3\mathrm{diff}\left(\mathrm{y1}\left(x\right)\,x\right)-\left(1-x^2\right)\mathrm{y1}\left(x\right)-x^2\mathrm{y2}\left(x\right)-\left(1+x+x^3\right)\mathrm{y3}\left(x\right)=0\,x\mathrm{diff}\left(\mathrm{y2}\left(x\right)\,x\right)-x^2\mathrm{y1}\left(x\right)-\left(1+x^3\right)\mathrm{y3}\left(x\right)=0\,x\mathrm{diff}\left(\mathrm{y3}\left(x\right)\,x\right)-x^3\mathrm{y1}\left(x\right)+x\mathrm{y2}\left(x\right)-\mathrm{y3}\left(x\right)=0\right]$

$\mathrm{sys}≔\left[x^3\left(\frac{\ⅆ}{\ⅆx}\mathrm{y1}\left(x\right)\right)-\left(-x^2+1\right)\mathrm{y1}\left(x\right)-x^2\mathrm{y2}\left(x\right)-\left(x^3+x+1\right)\mathrm{y3}\left(x\right)=0\,x\left(\frac{\ⅆ}{\ⅆx}\mathrm{y2}\left(x\right)\right)-x^2\mathrm{y1}\left(x\right)-\left(x^3+1\right)\mathrm{y3}\left(x\right)=0\,x\left(\frac{\ⅆ}{\ⅆx}\mathrm{y3}\left(x\right)\right)-x^3\mathrm{y1}\left(x\right)+x\mathrm{y2}\left(x\right)-\mathrm{y3}\left(x\right)=0\right]$

$\mathrm{vars}≔\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\,\mathrm{y3}\left(x\right)\right]\:$

$\mathrm{sol}≔\mathrm{RegularSolution}\left(\mathrm{sys}\,\mathrm{vars}\right)$

$\mathrm{sol}≔\left[\ln \left(x\right)\left(-x{\mathrm{\_c}}_1+\mathrm{O}\left(x^2\right)\right)-x{\mathrm{\_c}}_2+\mathrm{O}\left(x^2\right)\,\ln \left(x\right)\left(x{\mathrm{\_c}}_1+\mathrm{O}\left(x^2\right)\right)-{\mathrm{\_c}}_1+x\left({\mathrm{\_c}}_2-{\mathrm{\_c}}_1\right)+\mathrm{O}\left(x^2\right)\,\ln \left(x\right)\left(x{\mathrm{\_c}}_1+\mathrm{O}\left(x^2\right)\right)+x{\mathrm{\_c}}_2+\mathrm{O}\left(x^2\right)\right]$

$\mathrm{sol}≔\mathrm{ExtendRegularSolution}\left(\mathrm{sol}\,3\right)$

$\mathrm{sol}≔\left[\ln \left(x\right)\left(-x{\mathrm{\_c}}_1-\frac{9x^3{\mathrm{\_c}}_1}{4}+\mathrm{O}\left(x^4\right)\right)-x^2{\mathrm{\_c}}_1-x{\mathrm{\_c}}_2+x^3\left(-\frac{9{\mathrm{\_c}}_2}{4}-\frac{5{\mathrm{\_c}}_1}{4}\right)+\mathrm{O}\left(x^4\right)\,\ln \left(x\right)\left(x{\mathrm{\_c}}_1-\frac{x^2{\mathrm{\_c}}_1}{2}-\frac{x^3{\mathrm{\_c}}_1}{4}+\mathrm{O}\left(x^4\right)\right)-{\mathrm{\_c}}_1+x\left({\mathrm{\_c}}_2-{\mathrm{\_c}}_1\right)+x^2\left(-\frac{{\mathrm{\_c}}_2}{2}+\frac{5{\mathrm{\_c}}_1}{4}\right)+x^3\left(-\frac{{\mathrm{\_c}}_2}{4}-\frac{{\mathrm{\_c}}_1}{6}\right)+\mathrm{O}\left(x^4\right)\,\ln \left(x\right)\left(-x^2{\mathrm{\_c}}_1+x{\mathrm{\_c}}_1+\frac{x^3{\mathrm{\_c}}_1}{4}+\mathrm{O}\left(x^4\right)\right)+x{\mathrm{\_c}}_2+x^2\left(-{\mathrm{\_c}}_2+2{\mathrm{\_c}}_1\right)+x^3\left(\frac{{\mathrm{\_c}}_2}{4}-\frac{3{\mathrm{\_c}}_1}{4}\right)+\mathrm{O}\left(x^4\right)\right]$

$\mathrm{sol}≔\mathrm{ExtendRegularSolution}\left(\mathrm{sol}\,5\right)$

$\mathrm{sol}≔\left[\ln \left(x\right)\left(-x{\mathrm{\_c}}_1-\frac{9x^3{\mathrm{\_c}}_1}{4}-\frac{x^4{\mathrm{\_c}}_1}{2}-\frac{477x^5{\mathrm{\_c}}_1}{64}+\mathrm{O}\left(x^6\right)\right)-x^2{\mathrm{\_c}}_1-x{\mathrm{\_c}}_2+x^3\left(-\frac{9{\mathrm{\_c}}_2}{4}-\frac{5{\mathrm{\_c}}_1}{4}\right)+x^4\left(-\frac{{\mathrm{\_c}}_2}{2}-\frac{131{\mathrm{\_c}}_1}{36}\right)+x^5\left(-\frac{477{\mathrm{\_c}}_2}{64}-\frac{3475{\mathrm{\_c}}_1}{384}\right)+\mathrm{O}\left(x^6\right)\,\ln \left(x\right)\left(x{\mathrm{\_c}}_1-\frac{x^2{\mathrm{\_c}}_1}{2}-\frac{x^3{\mathrm{\_c}}_1}{4}+\frac{3x^4{\mathrm{\_c}}_1}{16}-\frac{211x^5{\mathrm{\_c}}_1}{320}+\mathrm{O}\left(x^6\right)\right)-{\mathrm{\_c}}_1+x\left({\mathrm{\_c}}_2-{\mathrm{\_c}}_1\right)+x^2\left(-\frac{{\mathrm{\_c}}_2}{2}+\frac{5{\mathrm{\_c}}_1}{4}\right)+x^3\left(-\frac{{\mathrm{\_c}}_2}{4}-\frac{{\mathrm{\_c}}_1}{6}\right)+x^4\left(\frac{3{\mathrm{\_c}}_2}{16}-\frac{151{\mathrm{\_c}}_1}{576}\right)+x^5\left(-\frac{211{\mathrm{\_c}}_2}{320}+\frac{7123{\mathrm{\_c}}_1}{28800}\right)+\mathrm{O}\left(x^6\right)\,\ln \left(x\right)\left(-x^2{\mathrm{\_c}}_1+x{\mathrm{\_c}}_1+\frac{x^3{\mathrm{\_c}}_1}{4}-\frac{x^4{\mathrm{\_c}}_1}{4}-\frac{3x^5{\mathrm{\_c}}_1}{64}+\mathrm{O}\left(x^6\right)\right)+x{\mathrm{\_c}}_2+x^2\left(-{\mathrm{\_c}}_2+2{\mathrm{\_c}}_1\right)+x^3\left(\frac{{\mathrm{\_c}}_2}{4}-\frac{3{\mathrm{\_c}}_1}{4}\right)+x^4\left(-\frac{{\mathrm{\_c}}_2}{4}+\frac{5{\mathrm{\_c}}_1}{36}\right)+x^5\left(-\frac{3{\mathrm{\_c}}_2}{64}-\frac{199{\mathrm{\_c}}_1}{1152}\right)+\mathrm{O}\left(x^6\right)\right]$

$A≔\mathrm{Matrix}\left(3\,3\,\left[\left[\frac{1-x^2}{x^3}\,\frac{1}{x}\,\frac{1+x+x^3}{x^3}\right]\,\left[x\,0\,\frac{1+x^3}{x}\right]\,\left[x^2\,-1\,\frac{1}{x}\right]\right]\right)$

$A≔\left[\begin{array}{ccc}\frac{-x^2+1}{x^3}& \frac{1}{x}& \frac{x^3+x+1}{x^3}\\ x& 0& \frac{x^3+1}{x}\\ x^2& \mathrm{-1}& \frac{1}{x}\end{array}\right]$

$\mathrm{sol}≔\mathrm{RegularSolution}\left(A\,x\,'\mathrm{differential}'\right)$

$\mathrm{sol}≔\left[\ln \left(x\right)\left(-x{\mathrm{\_c}}_1+\mathrm{O}\left(x^2\right)\right)-x{\mathrm{\_c}}_2+\mathrm{O}\left(x^2\right)\,\ln \left(x\right)\left(x{\mathrm{\_c}}_1+\mathrm{O}\left(x^2\right)\right)-{\mathrm{\_c}}_1+x\left({\mathrm{\_c}}_2-{\mathrm{\_c}}_1\right)+\mathrm{O}\left(x^2\right)\,\ln \left(x\right)\left(x{\mathrm{\_c}}_1+\mathrm{O}\left(x^2\right)\right)+x{\mathrm{\_c}}_2+\mathrm{O}\left(x^2\right)\right]$

$\mathrm{ExtendRegularSolution}\left(\mathrm{sol}\,3\right)$

$\left[\ln \left(x\right)\left(-x{\mathrm{\_c}}_1-\frac{9x^3{\mathrm{\_c}}_1}{4}+\mathrm{O}\left(x^4\right)\right)-x^2{\mathrm{\_c}}_1-x{\mathrm{\_c}}_2+x^3\left(-\frac{9{\mathrm{\_c}}_2}{4}-\frac{5{\mathrm{\_c}}_1}{4}\right)+\mathrm{O}\left(x^4\right)\,\ln \left(x\right)\left(x{\mathrm{\_c}}_1-\frac{x^2{\mathrm{\_c}}_1}{2}-\frac{x^3{\mathrm{\_c}}_1}{4}+\mathrm{O}\left(x^4\right)\right)-{\mathrm{\_c}}_1+x\left({\mathrm{\_c}}_2-{\mathrm{\_c}}_1\right)+x^2\left(-\frac{{\mathrm{\_c}}_2}{2}+\frac{5{\mathrm{\_c}}_1}{4}\right)+x^3\left(-\frac{{\mathrm{\_c}}_2}{4}-\frac{{\mathrm{\_c}}_1}{6}\right)+\mathrm{O}\left(x^4\right)\,\ln \left(x\right)\left(-x^2{\mathrm{\_c}}_1+x{\mathrm{\_c}}_1+\frac{x^3{\mathrm{\_c}}_1}{4}+\mathrm{O}\left(x^4\right)\right)+x{\mathrm{\_c}}_2+x^2\left(-{\mathrm{\_c}}_2+2{\mathrm{\_c}}_1\right)+x^3\left(\frac{{\mathrm{\_c}}_2}{4}-\frac{3{\mathrm{\_c}}_1}{4}\right)+\mathrm{O}\left(x^4\right)\right]$