\begin {align*} y^{\prime } & =F\left ( x,y\right ) \\ & =f\left ( x\right ) g\left ( y\right ) \end {align*}
The following flow chart gives the algorithm for solving separable ode.
ode internal name "separable"
Solved by separating and integrating. \(\frac {dy}{dx}\frac {1}{g\left ( y\right ) }=f\left ( x\right ) \). Integrating gives \(\int \frac {1}{g}dy=\int fdx\). If it is possible to do the integration of the LHS then explicit solution in \(y\) is obtained else the solution is implicit. The most difficult part is to determine that a given expression \(F\left ( x,y\right ) \) is separable or not. i.e. given \(y^{\prime }=F\left ( x,y\right ) \) to find \(f\left ( x\right ) \) and \(g\left ( y\right ) \,\). Code in solver is over 600 lines long just to determine this due to many edge cases.
Singular solutions are found by solving for \(y\) from \(g\left ( y\right ) =0\).