Liouville ODEs - Maple Help

Home : Support : Online Help : Mathematics : Differential Equations : Classifying ODEs : Second Order : odeadvisor/Liouville

Liouville ODEs

Description

 • The general form of the Liouville ODE is given by the following:
 > Liouville_ode := diff(y(x),x,x)+g(y(x))*diff(y(x),x)^2+f(x)*diff(y(x),x) = 0;
 ${\mathrm{Liouville_ode}}{:=}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){+}{g}{}\left({y}{}\left({x}\right)\right){}{\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right)}^{{2}}{+}{f}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){=}{0}$ (1)
 where g and f are arbitrary functions. See Goldstein and Braun, "Advanced Methods for the Solution of Differential Equations".

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor},\mathrm{symgen},\mathrm{symtest}\right):$
 > $\mathrm{odeadvisor}\left(\mathrm{Liouville_ode}\right)$
 $\left[{\mathrm{_Liouville}}{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_reducible}}{,}{\mathrm{_mu_x_y1}}\right]{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_reducible}}{,}{\mathrm{_mu_xy}}\right]\right]$ (2)

The Liouville ODE has the following symmetries (see dsolve,Lie):

 > $\mathrm{symmetries}≔\mathrm{symgen}\left(\mathrm{Liouville_ode}\right)$
 ${\mathrm{symmetries}}{:=}\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}{{ⅇ}}^{{-}\left({∫}{g}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\right)}\right]{,}\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}\left({∫}{{ⅇ}}^{{∫}{g}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\right){}{{ⅇ}}^{{-}\left({∫}{g}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\right)}\right]{,}\left[{\mathrm{_ξ}}{=}{{ⅇ}}^{{-}\left({∫}\left({-}{f}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}{,}{\mathrm{_η}}{=}{0}\right]{,}\left[{\mathrm{_ξ}}{=}\left({∫}{{ⅇ}}^{{∫}\left({-}{f}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){}{{ⅇ}}^{{-}\left({∫}\left({-}{f}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}{,}{\mathrm{_η}}{=}{0}\right]$ (3)

These symmetries can be tested using symtest

 > $\mathrm{map}\left(\mathrm{symtest},\left[\mathrm{symmetries}\right],\mathrm{Liouville_ode}\right)$
 $\left[{0}{,}{0}{,}{0}{,}{0}\right]$ (4)

Knowing two independent symmetries for a second order ODE almost always leads to its answer, as in the following Liouville ODE:

 > $\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{Liouville_ode}\right)$
 ${\mathrm{ans}}{:=}{{∫}}_{{}}^{{y}{}\left({x}\right)}{{ⅇ}}^{{∫}{g}{}\left({\mathrm{_b}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_b}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_b}}{-}{\mathrm{_C1}}{}\left({∫}{{ⅇ}}^{{-}\left({∫}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){-}{\mathrm{_C2}}{=}{0}$ (5)

Implicit and explicit answers for ODEs can be tested using odetest.

 > $\mathrm{odetest}\left(\mathrm{ans},\mathrm{Liouville_ode}\right)$
 ${0}$ (6)