The general form of a separable ODE is given by the following:

separable_ode := diff(y(x),x)=f(x)*g(y(x));

$\mathrm{separable\_ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=f\left(x\right)g\left(y\left(x\right)\right)$

where f(x) and g(y) are arbitrary functions. See Differentialgleichungen, by E. Kamke, p. 15. This type of ODE can be solved in a general manner by dsolve.

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

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

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

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

$\mathrm{dsolve}\left(\mathrm{separable\_ode}\right)$

$\int f\left(x\right)\ⅆx-\left({\int }_{\phantom{\mathrm{`\; `}}}^{y\left(x\right)}\frac{1}{g\left(\mathrm{\_a}\right)}\ⅆ\mathrm{\_a}\right)+\mathrm{c\_\_1}=0$