Differential Equation (Application)

The growth of a cell depends on the flow of nutrients through its surface.  Let W(t) be the weight of the cell at time t.  Assume that for a limited time the growth rate \displaystyle {dW}{dt} is proportional to W^{\frac{2}{3}}.  (If the density remains constant, then W is proportional to D^{3}, where D is the diameter of the cell, and the surface area is proportional to D^{2}, or equivalently W^{\frac{2}{3}).

Hence \displaystyle \frac{dW}{dt}=kW^{\frac{2}{3}} for some positive constant k.  Find the general solution to this differential equation.

Source: Rosenberg, R.L. (1984). Elementary Calculus: Course Notes. Ottawa, Canada: Holt, Rinehart and Winston. (Exercise 44: Question 4, p. 132)

Author: ascklee

Dr. Lee teaches at the Wee Kim Wee School of Communication and Information at the Nanyang Technological University in Singapore. He founded The Mathematics Digital Library in 2013.

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