Whereas most differential equations can only be solved by way of numerical approximation, some have a mathematically exact – analytical – solution.
The separation of variables lets us solve the differential equation of exponential growth analytically. The result is an exponential function.
The mathematical procedure to derive such a function necessitates a few not commonly familiar steps. Still most sources assume that everybody is an accomplished mathematician and skip the crucial details. But with the following work-up everything should become much clearer. If not, please let me know.
Separation of Variables (PDF, 14 pages)
Featured image: Colonies of the bacterium Legionella on a Petri dish. (Image credit: James Gathany. Centers for Disease Control and Prevention 2005. Public domain. https://phil.cdc.gov/Details.aspx?pid=7925, retrieved 6 January 2025)
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