On the analytical solution for the Pütter–Bertalanffy growth equation

• We developed the basic idea of allometric scaling in somatic growth dynamics.
• We proposed a standardized form of the Pütter–Bertalanffy equation.
• The analytical solution was successfully obtained.
• We discussed the mathematical behavior of the equation.

This study develops the basic idea of Pütter and Bertalanffy addressing the allometric scaling of anabolism and catabolism on somatic growth dynamics. We proposed a standardized form of the Pütter–Bertalanffy equation (PBE), which is given as the extended model of Richards function, and subsequently solved it. The analytical solution of the PBE was defined by an incomplete beta function and can take a wide range of shapes in its growth curve. The mathematical behavior of PBE due to the change in parameter values was briefly discussed. Most forms of solution consistently hold the implicit functional type with respect to the variable of body size.