symtest - Maple Help

DEtools

 symtest
 test a given symmetry

 Calling Sequence symtest(sym, ode, y(x))

Parameters

 sym - list of the coefficients of a symmetry generator ode - ordinary differential equation y(x) - (optional) indeterminate function of the ODE

Description

 • The symtest command checks whether a given pair of infinitesimals (coefficients of the symmetry generator) leave the given ODE invariant; that is, whether the given "symmetry" is actually a symmetry of the ODE. Similar to odetest, symtest returns $0$ when the result is valid or returns an algebraic expression obtained after simplifying the PDE for the infinitesimals associated with the given ODE.
 • If the result returned by symtest is not zero, the symmetry is not necessarily wrong. Sometimes, with further simplification, you can obtain the desired $0$ using commands such as expand, combine, and so on.
 • This function is part of the DEtools package, and so it can be used in the form symtest(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[symtest](..).

Examples

An ODE with an arbitrary function $F$

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{symgen},\mathrm{symtest}\right)$
 $\left[{\mathrm{symgen}}{,}{\mathrm{symtest}}\right]$ (1)
 > $\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right),x\right)=F\left(\frac{y\left(x\right)-x\mathrm{ln}\left(x\right)}{x}\right)+\mathrm{ln}\left(x\right)$
 ${\mathrm{ode}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{F}{}\left(\frac{{y}{}\left({x}\right){-}{x}{}{\mathrm{ln}}{}\left({x}\right)}{{x}}\right){+}{\mathrm{ln}}{}\left({x}\right)$ (2)

A pair of infinitesimals for the ODE above

 > $\mathrm{sym}≔\mathrm{symgen}\left(\mathrm{ode}\right)$
 ${\mathrm{sym}}{≔}\left[{\mathrm{_ξ}}{=}{x}{,}{\mathrm{_η}}{=}{x}{+}{y}\right]$ (3)

Testing these infinitesimals

 > $\mathrm{symtest}\left(\mathrm{sym},\mathrm{ode}\right)$
 ${0}$ (4)

A second order ODE

 > $\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right),x,x\right)=\frac{1}{{x}^{2}}\mathrm{_F1}\left(\frac{\mathrm{diff}\left(y\left(x\right),x\right)x}{y\left(x\right)}\right)y\left(x\right)$
 ${\mathrm{ode}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{\mathrm{_F1}}{}\left(\frac{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{x}}{{y}{}\left({x}\right)}\right){}{y}{}\left({x}\right)}{{{x}}^{{2}}}$ (5)
 > $\mathrm{sym}≔\mathrm{symgen}\left(\mathrm{ode}\right)$
 ${\mathrm{sym}}{≔}\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}{y}\right]{,}\left[{\mathrm{_ξ}}{=}{x}{,}{\mathrm{_η}}{=}{0}\right]$ (6)
 > $\mathrm{map}\left(\mathrm{symtest},\left[\mathrm{sym}\right],\mathrm{ode}\right)$
 $\left[{0}{,}{0}\right]$ (7)