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Nonhomogenous, Linear, Second– Outline Order, Differential ...

Nonhomogeneous, Linear, second - order , Differential EquationsOctober 4, 2017ME 501A Seminar in Engineering AnalysisPage 1 Nonhomogenous, Linear, second order , Differential EquationsLarry CarettoMechanical Engineering 501 ABSeminar in Engineering AnalysisOctober 4, 20172 Outline Review last class second - order nonhomogenous equations with constant coefficients Solution is sum of homogenous equation solution, yH, plus a particular solution, yP, for the nonhomogenous part Method of undetermined coefficients Variation of parameters3 Review y + y + y = 0 Three cases depending on 2= 2/4 Double root when = 2/4: y = (C1+ C2x) e x/2 Complex roots when > 2/4, 2> 0 y = e x/2 [Acos x+Bsin x] Distinct real roots when < 2/4 y = C1e 1x+ C2e 2x 2222244 Review y + y + y = 0 II Initial conditions y(0) = y0and y (0) = v0 Double root when = 2/4: y = [(v0 + y0 /2)x + y0] e x/2 Complex roots when > 2/4 y = e x/2 [y0cos x+ 1(v0 + y0 /2)sin x] Distinct real roots when < 2/4 y = C1e 1x+ C2e 2x C1= ( 2y0 v0)/( 2 ) C2= (v0 y0)/( 2 )5 Exam

Nonhomogeneous, Linear, Second-order, Differential Equations October 4, 2017 ME 501A Seminar in Engineering Analysis Page 2 Overdamping with 4km/c2 = 0.75-0.2

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