homogeneous - Maple Help

Solving Homogeneous ODEs of Class A

Description

 • The general form of the homogeneous equation of class A is given by the following:
 > homogeneousA_ode := diff(y(x),x)=f(y(x)/x);
 ${\mathrm{homogeneousA_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{f}{}\left(\frac{{y}{}\left({x}\right)}{{x}}\right)$ (1)
 where f(y(x)/x) is an arbitrary function. See Kamke's book, p. 19. This type of ODE can be solved in a general manner:

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (2)
 > $\mathrm{odeadvisor}\left(\mathrm{homogeneousA_ode}\right)$
 $\left[\left[{\mathrm{_homogeneous}}{,}{\mathrm{class A}}\right]{,}{\mathrm{_dAlembert}}\right]$ (3)
 > $\mathrm{dsolve}\left(\mathrm{homogeneousA_ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{RootOf}}{}\left({-}\left({{\int }}_{{}}^{{\mathrm{_Z}}}\frac{{1}}{{f}{}\left({\mathrm{_a}}\right){-}{\mathrm{_a}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}\right){+}{\mathrm{ln}}{}\left({x}\right){+}{\mathrm{_C1}}\right){}{x}$ (4)