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Generalized two-parameter equation of growth. (English) Zbl 0797.34006
The authors consider the differential equation \(y'= ay^ \alpha+ b\cdot y^ \beta\), \(y(t_ 0)= y_ 0\). Special cases are, with \(a,b> 0\), the Bertalanffy equation \(y'= a\cdot y^ \alpha- b\cdot y\), \(\alpha< 1\), the Gompertz equation \(y'= ay- by\log y\), \(\alpha= 1\), the logistic equation \(y'= ay-b\cdot y^ \alpha\), \(\alpha> 1\). They present analytical solutions for \(y_ 0\geq 0\) and \(\alpha< 1\), \(\alpha<\beta\) or \(\alpha> \beta\), \(\beta< 1\) and for \(y_ 0< 0\) and \(\alpha\), \(\beta\) integers. The solutions imply a slight generalization of the beta function and the exponential integral which result by simple integration of the transformed differential equations \((1- v)^ \delta v^{-1} v'= k\) or \(e^{\varepsilon v} v^{-1} v'= k\) with \(v\), \(k\), \(\delta\), \(\varepsilon\) appropriate. The behavior of the solution in dependence on the parameters is presented in tables. A possible application is to describe biological growth by a rate of increase and a rate of loss.
Reviewer: R.Repges (Aachen)

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34A05 Explicit solutions, first integrals of ordinary differential equations
92C15 Developmental biology, pattern formation
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