Solving Homogeneous ODEs of Class C - Maple Programming Help

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

Solving Homogeneous ODEs of Class C

Description

 • The general form of the homogeneous equation of class C is given by the following:
 > homogeneousC_ode := diff(y(x),x)=F((a*x+b*y(x)+c)/(r*x+s*y(x)+t));
 ${\mathrm{homogeneousC_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right){=}{F}{}\left(\frac{{a}{}{x}{+}{b}{}{y}{}\left({x}\right){+}{c}}{{r}{}{x}{+}{s}{}{y}{}\left({x}\right){+}{t}}\right)$ (1)
 where F is an arbitrary function of its argument. 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

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

A pair of infinitesimals for the homogeneousC_ode

 > $\mathrm{symgen}\left(\mathrm{homogeneousC_ode}\right)$
 $\left[{\mathrm{_ξ}}{=}\frac{{a}{}{s}{}{x}{-}{b}{}{r}{}{x}{-}{b}{}{t}{+}{c}{}{s}}{{a}{}{s}{-}{b}{}{r}}{,}{\mathrm{_η}}{=}\frac{{a}{}{s}{}{y}{-}{b}{}{r}{}{y}{+}{a}{}{t}{-}{c}{}{r}}{{a}{}{s}{-}{b}{}{r}}\right]$ (4)

The general solution for this ODE

 > $\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{homogeneousC_ode}\right)$
 ${\mathrm{ans}}{≔}{y}{}\left({x}\right){=}\frac{{a}{}{t}{-}{c}{}{r}{+}{\mathrm{RootOf}}{}\left({{∫}}_{{}}^{{\mathrm{_Z}}}\frac{{1}}{{F}{}\left(\frac{{\mathrm{_a}}{}{b}{-}{a}}{{\mathrm{_a}}{}{s}{-}{r}}\right){+}{\mathrm{_a}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}{\mathrm{ln}}{}\left({x}{}\left({a}{}{s}{-}{b}{}{r}\right){-}{b}{}{t}{+}{c}{}{s}\right){+}{\mathrm{_C1}}\right){}\left({x}{}\left({a}{}{s}{-}{b}{}{r}\right){-}{b}{}{t}{+}{c}{}{s}\right)}{{-}{a}{}{s}{+}{b}{}{r}}$ (5)

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

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