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Allometric growth in biology refers to relationships between sizes of parts of an organism (skull length and body length, for instance) If $ L_1(t) $ and $ L_2(t) $ are the sizes of two organs in an organism of age $ t, $ then $ L_1 $ and $ L_2 $ satisfy an allometric law if there specific growth rates are proportional:$ \frac {1}{L_1} \frac {dL_1}{dt} = k \frac {1}{L_2} \frac {dL_2}{dt} $where $ k $ is a constant.(a) Use the allometric law to write a differential equation relating $ L_1 $ and $ L_2 $ and solve it to express $ L_1 $ as a function of $ L_2. $(b) In a study of several species of unicellular algae, the proportionality constant in the allometric law relating $ B $ (cell biomass) and $ V $ (cell volume) was found to be $ k = 0.0794. $ Write $ B $ as a function of $ V. $

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a) $$L_{1}=K L_{2}^{k}, \text { where } K=e^{C}$$b) $B(V)=c V^{0.0794}$

Calculus 2 / BC

Chapter 9

Differential Equations

Section 3

Separable Equations

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

13:37

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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Allometric growth in biolo…

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Allometric Growth Suppose …

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