chp36_11 - PHY2061 R. D. Field Another Differential...

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PHY2061 R. D. Field Department of Physics chp36_11.doc University of Florida Another Differential Equation Consider the 2 nd order differential equation dxt dt D dx t Cx t 2 2 0 () ++ = , where C and D are constants. We solve this equation by turning it into an algebraic equation by looking for a solution of the form xt Ae at = . Substituting this into the differential equation yields, aD a C 2 0 = or a DD C =− ± 22 2 . Case I (C > (D/2) 2 , damped oscillations): For C > (D/2) 2 , i C D D i ± ± /( / ) / 2 2 ω , where CD (/ ) 2 2 , and the most general solution has the form: e Be e A t B t t t D t it Dt co s ( ) s in ( ) s ( ) s ( ) / / / / =+ = + = ′ +
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