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DEtools

 equinv
 look for the most general ODE invariant under a given symmetry

 Calling Sequence equinv([xi, eta], y(x)) equinv([[xi1, eta1], [xi2, eta2],...], y(x)) equinv([xi, eta], y(x), n) equinv([[xi1, eta1], [xi2, eta2],...], y(x), n)

Parameters

 [xi, eta] - list of the coefficients of the symmetry generator (infinitesimals) y(x) - indeterminate function to be used in the result n - positive integer, indicating that an nth order ODE is to be returned

Description

 • equinv takes as first argument either a list of two algebraic expressions, to be seen as the infinitesimals of a one-parameter Lie group, or a list of such pairs representing many different symmetry groups. It returns, within the possibilities of the system, the most general ODE simultaneously invariant under those symmetry groups. The second argument is the indeterminate function of that ODE.
 • If the number n is given as extra argument, equinv looks for an invariant ODE of order n; otherwise, it will look for a first order ODE.
 • As a rule, if we represent the infinitesimal generator as

$\mathrm{\xi }\left(x,y\right)\left(\frac{\partial }{\partial x}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right)+\mathrm{\eta }\left(x,y\right)\left(\frac{\partial }{\partial y}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right),$

 where x is the independent variable and y is the dependent one; equinv assumes that the order in which the infinitesimals xi and eta appear in the received list is $\left[\mathrm{\xi },\mathrm{\eta }\right]$.
 • In principle, there are no restrictions on the number of pairs of infinitesimals you can give at one time to equinv. However, the complexity of the problem increases with the number of pairs of infinitesimals given, and the problem may not have a solution. Also, when an nth order ODE is required (extra argument n) the given infinitesimals may contain _y1, _y2, ... which, as a convention, represent the derivative of the "indeterminate function" of the problem. In short, equinv also accepts dynamical symmetries as arguments.
 • equinv and buildsym may be useful in connection with the odeadvisor command and the $\mathrm{HINT}=\left[\mathrm{e1},\mathrm{e2}\right]$ option of dsolve, in order to study the relationship between symmetry patterns and ODE patterns (see dsolve/Lie).
 • This function is part of the DEtools package, and so it can be used in the form equinv(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[equinv](..).

Examples

Three pairs of infinitesimals not containing a 2-D subalgebra

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{sym}≔\left[\left[1,1\right],\left[x,y\right],\left[{x}^{2},{y}^{2}\right]\right]$
 ${\mathrm{sym}}{≔}\left[\left[{1}{,}{1}\right]{,}\left[{x}{,}{y}\right]{,}\left[{{x}}^{{2}}{,}{{y}}^{{2}}\right]\right]$ (1)

The most general second order ODE simultaneously invariant under these three point symmetries

 > $\mathrm{ODE2}≔\mathrm{equinv}\left(\mathrm{sym},y\left(x\right),2\right)$
 ${\mathrm{ODE2}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{3}}{{2}}}{}{\mathrm{_C1}}{+}{2}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{2}}{+}{2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)}{{-}{x}{+}{y}{}\left({x}\right)}$ (2)

The fact that ODE2 is invariant under the three pairs of infinitesimals above can be tested using the symtest command (see symtest):

 > $\mathrm{map}\left(\mathrm{symtest},\mathrm{sym},\mathrm{ODE2}\right)$
 $\left[{0}{,}{0}{,}{0}\right]$ (3)

Two pairs of dynamical symmetries

 > $\mathrm{dyn_sym}≔\left[\left[\mathrm{_y1},0\right],\left[0,\frac{1}{\mathrm{_y1}}\right]\right]$
 ${\mathrm{dyn_sym}}{≔}\left[\left[{\mathrm{_y1}}{,}{0}\right]{,}\left[{0}{,}\frac{{1}}{{\mathrm{_y1}}}\right]\right]$ (4)

The most general second order ODE simultaneously invariant under these two dynamical symmetries

 > $\mathrm{ODE2}≔\mathrm{equinv}\left(\mathrm{dyn_sym},y\left(x\right),2\right)$
 ${\mathrm{ODE2}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{2}}}{{\mathrm{_F1}}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{2}}{+}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{x}{-}{2}{}{y}{}\left({x}\right)}$ (5)
 > $\mathrm{map}\left(\mathrm{symtest},\mathrm{dyn_sym},\mathrm{ODE2}\right)$
 $\left[{0}{,}{0}\right]$ (6)