 Solving Homogeneous ODEs of Class D - Maple Programming Help

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Solving Homogeneous ODEs of Class D

Description

 • The general form of the homogeneous equation of class D is given by the following:
 > homogeneousD_ode := diff(y(x),x)= y(x)/x+g(x)*f(y(x)/x);
 ${\mathrm{homogeneousD_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{y}{}\left({x}\right)}{{x}}{+}{g}{}\left({x}\right){}{f}{}\left(\frac{{y}{}\left({x}\right)}{{x}}\right)$ (1)
 where f(y(x)/x) and g(x) are arbitrary functions of their arguments. See Differentialgleichungen, by E. Kamke, p. 20. 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

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

A pair of infinitesimals for homogeneousD_ode

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

The general solution for this ODE

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

Answers can be tested using odetest

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

Let's see how the answer above works when turning f into an explicit function; f is the identity mapping.

 > $f≔u↦u$
 ${f}{≔}{u}{↦}{u}$ (7)
 > $\mathrm{allvalues}\left(\mathrm{value}\left(\mathrm{ans}\right)\right)$
 ${y}{}\left({x}\right){=}{{ⅇ}}^{{\int }\frac{{g}{}\left({x}\right)}{{x}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{\mathrm{_C1}}}{}{x}$ (8)
 > $\mathrm{odetest}\left(\mathrm{ans},\mathrm{homogeneousD_ode}\right)$
 ${0}$ (9)