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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)

##### MSC:
 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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