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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 γv, where m > 0 is the mass of the object, g is the acceleration due to gravity near sea level, and γ > 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 = γ/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 lefthand 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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 Spring '11
 Smith
 Equations, Constant of integration, Equations Analytic Solutions

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