Solving Homogeneous ODEs of Class G - Maple Help

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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);
 ${\mathrm{homogeneousG_ode}}{:=}\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right){=}\frac{{y}{}\left({x}\right){}{F}{}\left(\frac{{y}{}\left({x}\right)}{{{x}}^{{\mathrm{α}}}}\right)}{{x}}$ (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

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

The general solution for this ODE

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

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

 > $\mathrm{odetest}\left(\mathrm{ans},\mathrm{homogeneousG_ode}\right)$
 ${0}$ (6)
 See Also DEtools, odeadvisor, dsolve, and ?odeadvisor, where is one of: quadrature, linear, separable, Bernoulli, exact, homogeneous, homogeneousB, homogeneousC, homogeneousD, homogeneousG, Chini, Riccati, Abel, Abel2A, Abel2C, rational, Clairaut, dAlembert, sym_implicit, patterns; for other differential orders see odeadvisor,types.