The sym_implicit subroutine of the odeadvisor command tests if a given first order ODE in "implicit form" (that is, dy/dx cannot be isolated) has one or more of the following symmetries:

[xi=0, eta=y], [xi=0, eta=x], [xi=0, eta=1/x], [xi=0, eta=1/y],
[xi=x, eta=0], [xi=y, eta=0], [xi=1/x, eta=0], [xi=1/y, eta=0],
[xi=x, eta=y], [xi=y, eta=x]:

where the infinitesimal symmetry generator is given by the following:

G := f -> xi*diff(f,x) + eta*diff(f,y);

$G≔f\→\mathrm{\ξ}\left(\frac{\partial }{\partial x}f\right)\+\mathrm{\η}\left(\frac{\partial }{\partial y}f\right)$

This routine is relevant when using symmetry methods for solving high-degree ODEs (nonlinear in dy/dx). The cases [xi=0, eta=y], [xi=1/x, eta=y] and [xi=x, eta=y] cover the families of homogeneous ODEs mentioned in Murphy's book, pages 63-64.

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

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

$X≔\left[\frac{1}{y}\,0\right]$

$X≔\left[\frac{1}{y}\,0\right]$

$\mathrm{implicit\_ode}≔F\left(y\left(x\right)\,\frac{1}{\mathrm{diff}\left(y\left(x\right)\,x\right)}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)x+y\left(x\right)\right)\right)=0$

$\mathrm{implicit\_ode}≔F\left(y\left(x\right)\,\frac{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)x+y\left(x\right)}{\frac{\ⅆ}{\ⅆx}y\left(x\right)}\right)=0$

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

$\left[\left[\mathrm{\_1st\_order}\,\mathrm{\_with\_symmetry\_[F(x)*G(y),0]}\right]\right]$

$\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{implicit\_ode}\right)$

$\mathrm{ans}≔xy\left(x\right)-\left({\int }_{\phantom{\mathrm{`\; `}}}^{y\left(x\right)}\mathrm{RootOf}\left(F\left(\mathrm{\_a}\,\mathrm{\_Z}\right)\right)\ⅆ\mathrm{\_a}\right)-\mathrm{c\_\_1}=0$

$\mathrm{odetest}\left(\mathrm{ans}\,\mathrm{implicit\_ode}\right)$

$0$