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

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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 d x t dt D dx t dt 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 x t Ae at ( ) = . Substituting this into the differential equation yields, a Da C 2 0 + + = or a D D C = − ± 2 2 2 . Case I (C > (D/2) 2 , damped oscillations): For C > (D/2) 2 , a D i C D D i = − ± = − ± / ( / ) / 2 2 2 2 ω , where ′ = ω C D ( / ) 2 2 , and the most general solution has the form: ( ) ( ) x t e Ae Be x t e A t B t x t Ae t x t Ae t Dt i t i
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