See Differentialgleichungen, by E. Kamke, p. 30. The technique consists mainly of looking for a parametric solution. Consider, for instance, the case y=G(x,diff(y,x)).

ode := y=G(x,diff(y(x),x));

Choosing the parametrization

para := diff(y(x),x) = t;

ode1 := subs(para,x=x(t),y=y(t), ode);

From the equations above and using the chain rule  , it is possible to obtain another ODE for  as follows:

ode_draft := diff(x(t),t) = diff(rhs(ode1),t)/t:

ode2 := diff(x(t),t)=solve(ode_draft,diff(x(t),t));

You should therefore solve  for  and determine  by introducing the resulting  in . Note that, when G does not depend on  is a quadrature. ODEs matching the pattern  are solved using the same ideas, and ODEs matching the patterns , or  are just particular cases.

Although any ODE can be attempted using the scheme outlined above, generally speaking, there are four cases which can be better dealt with by looking for a parametric solution; they are:

   (particular case)

   (particular case)

Parametric solutions are available by giving the optional argument 'parametric' to dsolve. By default, when the ODE is of high degree in , dsolve tries the parametric scheme, along with a set of related methods for this type of ODE. However, this scheme may also be of help in some cases in which  can be isolated.