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DEtools

 kovacicsols
 find solutions of a second order linear ODE

 Calling Sequence kovacicsols(lode, v) kovacicsols(coeff_list, x)

Parameters

 lode - second order linear differential equation with rational coefficients v - dependent variable of the lode coeff_list - list of coefficients of a linear ode x - independent variable of the lode

Description

 • The kovacicsols routine returns a list of Liouvillian solutions of a second order linear differential equation with rational coefficients. It uses the algorithm of Kovacic to determine if such solutions exist, and if so, to determine them.
 Functions that can be expressed in terms of exp, int, and algebraic functions, are called Liouvillian functions. The typical example is ${ⅇ}^{{\int }R\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x}$ where $R\left(x\right)$ is an algebraic function, that is, a function that can be expressed in the form $\mathrm{RootOf}\left(F\left(x,y\right),y\right)$ for some polynomial $F\left(x,y\right)$.
 • By default, kovacicsols currently returns answers in a form that is different from the regular Kovacic algorithm; if the projective differential Galois group is A4, S4, or A5, then kovacicsols presents the answer in terms of hypergeometric functions. This is done by reducing the equation to an equation with only 3 singularities using the formulas given in  (see references at the end).
 To change this behavior, set _Env_kovacicsols_pullback := false. Then, kovacicsols presents the solutions in the same form as the usual Kovacic algorithm.
 • There are two input forms. The first has as the first argument a linear differential equation in diff or D form and as the second argument the variable in the differential equation.
 • A second input sequence accepts for the first argument a list of coefficients of the linear ode, and for the second argument the independent variable of the lode. This input sequence is useful for programming with the kovacicsols routine.
 • In the second calling sequence, the list of coefficients is given in order from low differential order to high differential order and does not include the nonhomogeneous term.  The coefficients must be rational.
 • This function is part of the DEtools package, and so it can be used in the form kovacicsols(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[kovacicsols](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{ode}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)+\frac{3\left({x}^{2}-x+1\right)y\left(x\right)}{16{\left(x-1\right)}^{2}{x}^{2}}:$
 > $\mathrm{kovacicsols}\left(\mathrm{ode},y\left(x\right)\right)$
 $\left[\sqrt{{x}{-}{1}}{}{\left(\frac{{-}{{x}}^{{3}}{{2}}}{+}{x}}{{-}\sqrt{{x}}{-}{1}}\right)}^{{1}}{{4}}}{,}\sqrt{{x}{-}{1}}{}{\left(\frac{{{x}}^{{3}}{{2}}}{+}{x}}{{-}{1}{+}\sqrt{{x}}}\right)}^{{1}}{{4}}}\right]$ (1)
 > $A≔\left[3-12{x}^{2},-8{x}^{3}+5x-1,{x}^{2}-{x}^{4}\right]:$
 > $\mathrm{kovacicsols}\left(A,x\right)$
 $\left[\frac{{{ⅇ}}^{{-}\frac{{1}}{{x}}}}{{{x}}^{{3}}{}\left({x}{-}{1}\right)}{,}\frac{{{ⅇ}}^{{-}\frac{{1}}{{x}}}{}\left({{ⅇ}}^{{-1}}{}{{\mathrm{Ei}}}_{{1}}{}\left({-}\frac{{x}{+}{1}}{{x}}\right){+}{x}{}{{ⅇ}}^{\frac{{1}}{{x}}}\right)}{{{x}}^{{3}}{}\left({x}{-}{1}\right)}\right]$ (2)
 > $\mathrm{ode}≔48x\left(x+1\right)\left(5x-4\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)+8\left(25x+16\right)\left(x-2\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)-\left(5x+68\right)y\left(x\right)$
 ${\mathrm{ode}}{≔}{48}{}{x}{}\left({x}{+}{1}\right){}\left({5}{}{x}{-}{4}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{8}{}\left({25}{}{x}{+}{16}\right){}\left({x}{-}{2}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}\left({5}{}{x}{+}{68}\right){}{y}{}\left({x}\right)$ (3)

Compute a solution in terms of the hypergeometric function.

 > ${\mathrm{kovacicsols}\left(\mathrm{ode},y\left(x\right)\right)}_{1}$
 ${\left(\frac{{{x}}^{{2}}{-}{12}{}{x}{-}{16}}{{x}}\right)}^{{1}}{{4}}}{}{\mathrm{hypergeom}}{}\left(\left[{-}\frac{{1}}{{12}}{,}\frac{{1}}{{4}}\right]{,}\left[\frac{{2}}{{3}}\right]{,}{-}\frac{{\left({5}{}{{x}}^{{2}}{+}{24}{}{x}{+}{16}\right)}^{{3}}}{{x}{}{\left({{x}}^{{2}}{-}{12}{}{x}{-}{16}\right)}^{{3}}}\right)$ (4)

Compute a solution in terms of RootOfs and radicals.

 > $\mathrm{_Env_kovacicsols_pullback}≔\mathrm{false}$
 ${\mathrm{_Env_kovacicsols_pullback}}{≔}{\mathrm{false}}$ (5)
 > ${\mathrm{kovacicsols}\left(\mathrm{ode},y\left(x\right)\right)}_{1}$
 $\frac{\sqrt{\frac{\left({x}{+}{2}{}{{x}}^{{2}}{{3}}}{-}{4}{}{{x}}^{{1}}{{3}}}{+}{4}\right){}\left({-}{6}{}\sqrt{{3}}{}\sqrt{{{x}}^{{2}}{{3}}}{-}{{x}}^{{1}}{{3}}}{+}{1}}{}{{x}}^{{1}}{{3}}}{+}{6}{}\sqrt{{3}}{}{{x}}^{{2}}{{3}}}{-}{4}{}\sqrt{{{x}}^{{2}}{{3}}}{-}{{x}}^{{1}}{{3}}}{+}{1}}{}{{x}}^{{2}}{{3}}}{+}{4}{}{x}{+}{2}{}\sqrt{{3}}{}\sqrt{{{x}}^{{2}}{{3}}}{-}{{x}}^{{1}}{{3}}}{+}{1}}{-}{5}{}\sqrt{{3}}{}{{x}}^{{1}}{{3}}}{-}{3}{}\sqrt{{{x}}^{{2}}{{3}}}{-}{{x}}^{{1}}{{3}}}{+}{1}}{}{{x}}^{{1}}{{3}}}{+}{{x}}^{{2}}{{3}}}{+}{4}{}\sqrt{{3}}{-}{2}{}\sqrt{{{x}}^{{2}}{{3}}}{-}{{x}}^{{1}}{{3}}}{+}{1}}{+}{2}{}{{x}}^{{1}}{{3}}}{+}{2}\right)}{{6}{}\sqrt{{3}}{}\sqrt{{{x}}^{{2}}{{3}}}{-}{{x}}^{{1}}{{3}}}{+}{1}}{}{{x}}^{{1}}{{3}}}{-}{6}{}\sqrt{{3}}{}{{x}}^{{2}}{{3}}}{-}{4}{}\sqrt{{{x}}^{{2}}{{3}}}{-}{{x}}^{{1}}{{3}}}{+}{1}}{}{{x}}^{{2}}{{3}}}{+}{4}{}{x}{-}{2}{}\sqrt{{3}}{}\sqrt{{{x}}^{{2}}{{3}}}{-}{{x}}^{{1}}{{3}}}{+}{1}}{+}{5}{}\sqrt{{3}}{}{{x}}^{{1}}{{3}}}{-}{3}{}\sqrt{{{x}}^{{2}}{{3}}}{-}{{x}}^{{1}}{{3}}}{+}{1}}{}{{x}}^{{1}}{{3}}}{+}{{x}}^{{2}}{{3}}}{-}{4}{}\sqrt{{3}}{-}{2}{}\sqrt{{{x}}^{{2}}{{3}}}{-}{{x}}^{{1}}{{3}}}{+}{1}}{+}{2}{}{{x}}^{{1}}{{3}}}{+}{2}}}{}{\left({{x}}^{{1}}{{3}}}{+}{1}\right)}^{{1}}{{4}}}{}{\left(\frac{\sqrt{{3}}{}{{x}}^{{1}}{{3}}}{-}\sqrt{{3}}{-}{2}{}\sqrt{{{x}}^{{2}}{{3}}}{-}{{x}}^{{1}}{{3}}}{+}{1}}}{\sqrt{{3}}{}{{x}}^{{1}}{{3}}}{-}\sqrt{{3}}{+}{2}{}\sqrt{{{x}}^{{2}}{{3}}}{-}{{x}}^{{1}}{{3}}}{+}{1}}}\right)}^{{1}}{{8}}}}{{{x}}^{{1}}{{3}}}}$ (6)

References

 Kovacic, J. "An algorithm for solving second order linear homogeneous equations". J. Symb. Comp. Vol. 2. (1986): 3-43.
 Weil, J.A. "Recent Algorithms for Solving Second-Order Differential Equations", The Algorithm Project, http://pauillac.inria.fr/algo/seminars/sem01-02/weil.pdf; accessed 2002.