1) f0(x) = f1(x)*g(x)-f2(x)*g(x)^2

This case can be solved as follows:

2) Another case which can be solved:

f1(x) = 2*f2(x)*g(x)-diff(g(x),x)

Although the answer for this case can be obtained using standard methods (an integrating factor is easily found), the use of symmetry methods can provide an explicit solution. The infinitesimals for this case are given by

To indicate the use of symmetry methods "at first", we can explicitly indicate an integration method (see dsolve); for instance, to use the canonical coordinates of the invariance group:

DEtools, dsolve, odeadvisor, and ?odeadvisor,<TYPE> where <TYPE> is one of: quadrature, linear, separable, Bernoulli, exact, homogeneous, homogeneousB, homogeneousC, homogeneousD, homogeneousG, Chini, Riccati, Abel, Abel2C, rational, Clairaut, dAlembert, sym_implicit, patterns; for other differential orders see odeadvisor,TYPES.