Differential Equations

Verify that the differential equation for logistic growth \displaystyle \frac{dN}{dt}=cN\left(K-N\right) can be converted into the linear differential equation \displaystyle \frac{dx}{dt}+cKx=c by replacing N by \displaystyle \frac{1}{x}.

    \[x=\frac{1}{N}\]

    \[\frac{dx}{dt}=-\frac{1}{N^{2}}\frac{dN}{dt}\]

Obtain the equation of logistic growth by first solving the differential equation for x a function of t and then reintroducing N.

Source: R.L. Rosenberg. (1984). Elementary Calculus: Course Notes. Ottawa, Canada: Holt, Rinehart and Winston. (Exercise 45: Question 3, p. 134)

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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