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Unformatted text preview: Math 334 Lecture #2 1.2: Solutions of Some Differential Equations Analytic Solutions of a Simple Differential Equation. Recall that the differen- tial equation that models an object falling in the atmosphere near sea level is m dv dt = mg- v, where m > 0 is the mass of the object, g is the acceleration due to gravity near sea level, and > 0 is the drag coefficient. This differential equation is of the form dy dt = ay- b with a 6 = 0. [For the falling object model, a =- /m and b =- g .] If y 6 = b/a [which is the equilibrium solution], then the differential equation can be written in the form 1 y- b/a dy dt = a. [Question: What does the left-hand side remind you of? The chain rule applied to ln | y- b/a | .] Integrating both sides with respect to t gives ln | y- b/a | = at + C, where C is the arbitrary constant of integration. [Integration on each side introduces an arbitrary constant there; these combine into one arbitrary constant.] At this point the differential equation is solved except that...
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.
- Spring '11