Solving d'Alembert ODEs
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));
dAlembert_ode ≔ y⁡x=x⁢f⁡ⅆⅆx⁢y⁡x+g⁡ⅆⅆx⁢y⁡x
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].
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:
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