The general form of the exact ODE is given by:

exact_ode := D[2](C)(x,y(x))*diff(y(x),x)+D[1](C)(x,y(x))=0;

$\mathrm{exact\_ode}≔{\mathrm{D}}_2\left(C\right)\left(x\,y\left(x\right)\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+{\mathrm{D}}_1\left(C\right)\left(x\,y\left(x\right)\right)=0$

where C is an arbitrary function of its arguments. See Kamke's book, p. 28. This type of ODE can be solved in a general manner by dsolve, and the infinitesimals can also be determined by symgen.

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

$\left[\mathrm{odeadvisor}\,\mathrm{symgen}\,\mathrm{symtest}\right]$

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

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

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

$\mathrm{ans}≔C\left(x\,y\left(x\right)\right)+\mathrm{c\_\_1}=0$

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

$0$

$\mathrm{sym}≔\mathrm{symgen}\left(\mathrm{exact\_ode}\right)$

$\mathrm{sym}≔\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=\frac{1}{\frac{\partial }{\partial y}C\left(x\,y\right)}\right]$

$\mathrm{symtest}\left(\mathrm{sym}\,\mathrm{exact\_ode}\right)$

$0$