Solving Homogeneous ODEs of Class B - Maple Programming Help

Home : Support : Online Help : Mathematics : Differential Equations : Classifying ODEs : First Order : odeadvisor/homogeneousB

Solving Homogeneous ODEs of Class B

Description

 • The general form of the homogeneous equation of class B is given by the following:
 > homogeneousB_ode := F(diff(y(x),x), y(x)/x);
 ${\mathrm{homogeneousB_ode}}{≔}{F}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right){,}\frac{{y}{}\left({x}\right)}{{x}}\right)$ (1)
 where F is an arbitrary functions of its arguments. See Differentialgleichungen, by E. Kamke, p. 19. 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

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{symgen}\right)$
 $\left[{\mathrm{symgen}}\right]$ (2)

A pair of infinitesimals for homogeneousB_ode

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

The general solution for this ODE

 > $\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{homogeneousB_ode}\right)$
 ${\mathrm{ans}}{≔}{y}{}\left({x}\right){=}{\mathrm{RootOf}}{}\left({-}\left({{∫}}_{{}}^{{\mathrm{_Z}}}\frac{{1}}{{\mathrm{RootOf}}{}\left({F}{}\left({\mathrm{_Z}}{,}{\mathrm{_a}}\right)\right){-}{\mathrm{_a}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}\right){+}{\mathrm{ln}}{}\left({x}\right){+}{\mathrm{_C1}}\right){}{x}$ (4)

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

 > $\mathrm{odetest}\left(\mathrm{ans},\mathrm{homogeneousB_ode}\right)$
 ${0}$ (5)