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));

$\mathrm{dAlembert\_ode}≔y\left(x\right)=xf\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+g\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)$

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].

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\right)$

$\left[\mathrm{odeadvisor}\right]$

$\mathrm{odeadvisor}\left(\mathrm{dAlembert\_ode}\right)$

$\left[\mathrm{\_dAlembert}\right]$

$\mathrm{dsolve}\left(\mathrm{dAlembert\_ode}\right)$

$y\left(x\right)=x\mathrm{RootOf}\left(\mathrm{\_Z}-f\left(\mathrm{\_Z}\right)\right)+g\left(\mathrm{RootOf}\left(\mathrm{\_Z}-f\left(\mathrm{\_Z}\right)\right)\right),\left[x\left(\mathrm{\_T}\right)={\ⅇ}^{\int \frac{\frac{\ⅆ}{\ⅆ\mathrm{\_T}}f\left(\mathrm{\_T}\right)}{\mathrm{\_T}-f\left(\mathrm{\_T}\right)}\ⅆ\mathrm{\_T}}\left(\int \frac{\left(\frac{\ⅆ}{\ⅆ\mathrm{\_T}}g\left(\mathrm{\_T}\right)\right){\ⅇ}^{-\left(\int \frac{\frac{\ⅆ}{\ⅆ\mathrm{\_T}}f\left(\mathrm{\_T}\right)}{\mathrm{\_T}-f\left(\mathrm{\_T}\right)}\ⅆ\mathrm{\_T}\right)}}{\mathrm{\_T}-f\left(\mathrm{\_T}\right)}\ⅆ\mathrm{\_T}+\mathrm{c\_\_1}\right)\,y\left(\mathrm{\_T}\right)={\ⅇ}^{\int \frac{\frac{\ⅆ}{\ⅆ\mathrm{\_T}}f\left(\mathrm{\_T}\right)}{\mathrm{\_T}-f\left(\mathrm{\_T}\right)}\ⅆ\mathrm{\_T}}\left(\int \frac{\left(\frac{\ⅆ}{\ⅆ\mathrm{\_T}}g\left(\mathrm{\_T}\right)\right){\ⅇ}^{-\left(\int \frac{\frac{\ⅆ}{\ⅆ\mathrm{\_T}}f\left(\mathrm{\_T}\right)}{\mathrm{\_T}-f\left(\mathrm{\_T}\right)}\ⅆ\mathrm{\_T}\right)}}{\mathrm{\_T}-f\left(\mathrm{\_T}\right)}\ⅆ\mathrm{\_T}+\mathrm{c\_\_1}\right)f\left(\mathrm{\_T}\right)+g\left(\mathrm{\_T}\right)\right]$