Numerical investigation of a generalized class of Abel-type differential equations: Methodological comparison based on parameter parity

We study the Abel-type family \(y'=C\,y^r(1-y)^s\) under a parity-driven mapping of \((r,s)\), which yields symmetric dynamics for odd \(k\) and asymmetric, potentially stiff dynamics for even \(k\). We correct the normalization by peaking at the true maximizer \(y^=r/(r+s)\) and provide the analytic Jacobian \(g'(y)\) for implicit solvers. A matched-accuracy benchmarking protocol sweeps rtol/atol and reports global errors against ultra-tight references (separable/explicit for odd \(k\), Radau for even \(k\)), alongside wall time, \(nfev\), \(njev\), linear-solve counts, rejected steps, and step-size histories. Stiffness is quantified through the proxy \(τ(t)=1/ g'(y(t))\) and correlated with step-size adaptation; trajectories are constrained to \(y∈[0,1]\) via terminal events. Across tolerances, DOP853 and LSODA are strong all-rounders in non-stiff regimes, while Radau/BDF dominate when asymmetry and proximity to multiple roots induce stiffness; observed orders align with nominal ones under matched error. The study clarifies how parity and nonlinearity govern solver efficiency for polynomial nonlinearities and provides full environment details and code for reproducibility.