homogeneousG - Maple Help

Solving Homogeneous ODEs of Class G

Description

 • The general form of the homogeneous equation of class G is given by the following:
 > homogeneousG_ode := diff(y(x),x) = y(x)/x*F(y(x)/x^alpha);
 ${\mathrm{homogeneousG_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{y}{}\left({x}\right){}{F}{}\left(\frac{{y}{}\left({x}\right)}{{{x}}^{{\mathrm{\alpha }}}}\right)}{{x}}$ (1)
 where F is an arbitrary functions of its argument. This type of ODE can be solved in a general manner by dsolve and the coefficients of the infinitesimal symmetry generator are also found by symgen.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor},\mathrm{symgen}\right)$
 $\left[{\mathrm{odeadvisor}}{,}{\mathrm{symgen}}\right]$ (2)
 > $\mathrm{odeadvisor}\left(\mathrm{homogeneousG_ode}\right)$
 $\left[\left[{\mathrm{_homogeneous}}{,}{\mathrm{class G}}\right]\right]$ (3)

A pair of infinitesimals for the homogeneousG_ode

 > $\mathrm{symgen}\left(\mathrm{homogeneousG_ode}\right)$
 $\left[{\mathrm{_ξ}}{=}{x}{,}{\mathrm{_η}}{=}{\mathrm{\alpha }}{}{y}\right]$ (4)

The general solution for this ODE

 > $\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{homogeneousG_ode}\right)$
 ${\mathrm{ans}}{≔}{y}{}\left({x}\right){=}{\mathrm{RootOf}}{}\left({-}{\mathrm{ln}}{}\left({x}\right){+}{\mathrm{_C1}}{+}{{\int }}_{{}}^{{\mathrm{_Z}}}\frac{{1}}{{\mathrm{_a}}{}\left({-}{\mathrm{\alpha }}{+}{F}{}\left({\mathrm{_a}}\right)\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}\right){}{{x}}^{{\mathrm{\alpha }}}$ (5)

Explicit or implicit results can be tested, in principle, using odetest

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