gensys(ODE, [_xi = f(x, y, ..), _eta = g(x, y, ..)], y(x))

gensys(ODE, _mu = f(x, y, ..), y(x))

1. Determine if an integrating factor of the form  exists for the second order nonlinear ODE

and if so, determine the integrating factor itself.

Set up the determining PDE system for these integrating factors (gensys enters in this step), then try to simplify it (perhaps using casesplit), then try to solve it. The determining PDE system is set up as follows:

This system (each equation above is assumed to be equal to zero) has four equations,

and by taking into account their integrability conditions it is simplified to

Thus, mu does not depend on  and the remaining equation leads to

Note that this result can be tested using mutest

and can be used to obtain a first integral (that is, to reduce the order of ODE).

First integrals can be tested using firtest.

2. Determine if an integrating factor of the form  exists for the same second order ODE.

So this ODE has no integrating factor of the form .

3. Determine whether or not this ODE has point symmetries.

Set the determining system for the coefficients of the infinitesimal symmetry generator (so-called "infinitesimals") as follows:

(in above, [_xi = xi(x,y), _eta = eta(x,y)] also works). This system consists of 4 equations,

and in simplifying these equations by taking into account their integrability conditions one obtains

showing that this ODE has no point symmetries.

DEtools[buildsym], DEtools[equinv], DEtools[firint], DEtools[firtest], DEtools[intfactor], DEtools[Lie], DEtools[mutest], DEtools[odepde], DEtools[symgen], dsolve[Lie], PDEtools, PDEtools[casesplit], pdsolve