Solving d'Alembert ODEs - Maple Programming Help

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

Solving d'Alembert ODEs

Description

 • The general form of the d'Alembert ODE is given by:
 > dAlembert_ode := y(x)=x*f(diff(y(x),x))+g(diff(y(x),x));
 ${\mathrm{dAlembert_ode}}{≔}{y}{}\left({x}\right){=}{x}{}{f}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{g}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (1)
 where f and g are arbitrary functions. See Differentialgleichungen, by E. Kamke, p. 31. This ODE is actually a generalization of the Clairaut ODE, and is almost always dealt with by looking for a solution in parametric form. For more information, see odeadvisor[patterns].

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (2)
 > $\mathrm{odeadvisor}\left(\mathrm{dAlembert_ode}\right)$
 $\left[{\mathrm{_dAlembert}}\right]$ (3)

The general form of the solution for the d'Alembert ODE is returned by dsolve in parametric form, together with a possible singular solution, as follows:

 > $\mathrm{dsolve}\left(\mathrm{dAlembert_ode}\right)$
 ${y}{}\left({x}\right){=}{x}{}{\mathrm{RootOf}}{}\left({-}{f}{}\left({\mathrm{_Z}}\right){+}{\mathrm{_Z}}\right){+}{g}{}\left({\mathrm{RootOf}}{}\left({-}{f}{}\left({\mathrm{_Z}}\right){+}{\mathrm{_Z}}\right)\right){,}\left[{x}{}\left({\mathrm{_T}}\right){=}{{ⅇ}}^{{\int }\frac{\frac{{ⅆ}}{{ⅆ}{\mathrm{_T}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({\mathrm{_T}}\right)}{{\mathrm{_T}}{-}{f}{}\left({\mathrm{_T}}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_T}}}{}\left({\int }\frac{\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{_T}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({\mathrm{_T}}\right)\right){}{{ⅇ}}^{{-}\left({\int }\frac{\frac{{ⅆ}}{{ⅆ}{\mathrm{_T}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({\mathrm{_T}}\right)}{{\mathrm{_T}}{-}{f}{}\left({\mathrm{_T}}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_T}}\right)}}{{\mathrm{_T}}{-}{f}{}\left({\mathrm{_T}}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_T}}{+}{\mathrm{_C1}}\right){,}{y}{}\left({\mathrm{_T}}\right){=}{{ⅇ}}^{{\int }\frac{\frac{{ⅆ}}{{ⅆ}{\mathrm{_T}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({\mathrm{_T}}\right)}{{\mathrm{_T}}{-}{f}{}\left({\mathrm{_T}}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_T}}}{}\left({\int }\frac{\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{_T}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({\mathrm{_T}}\right)\right){}{{ⅇ}}^{{-}\left({\int }\frac{\frac{{ⅆ}}{{ⅆ}{\mathrm{_T}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({\mathrm{_T}}\right)}{{\mathrm{_T}}{-}{f}{}\left({\mathrm{_T}}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_T}}\right)}}{{\mathrm{_T}}{-}{f}{}\left({\mathrm{_T}}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_T}}{+}{\mathrm{_C1}}\right){}{f}{}\left({\mathrm{_T}}\right){+}{g}{}\left({\mathrm{_T}}\right)\right]$ (4)