A certain spring-mass system satisfies the initial value problem u" + 2u' + 10u = kg(t), u(0) = 0, u'(0) = 0, where g(t) = uz(t) – us (t) and k > 0 is a parameter. a) Sketch the graph of the forcing...

0 is a parameter. a) Sketch the graph of the forcing function on an appropriate interval. 0.8 0.6 0.4 0.2 1 4 b) Solve the initial value problem. NOTE: Denote the two Heaviside functions by uc(t) and ua(t). Indicate separately the values of c and d knowing that c is the minimum of the two values. u(t) = "/>
Extracted text: A certain spring-mass system satisfies the initial value problem u" + 2u' + 10u = kg(t), u(0) = 0, u'(0) = 0, where g(t) = uz(t) – us (t) and k > 0 is a parameter. a) Sketch the graph of the forcing function on an appropriate interval. 0.8 0.6 0.4 0.2 1 4 b) Solve the initial value problem. NOTE: Denote the two Heaviside functions by uc(t) and ua(t). Indicate separately the values of c and d knowing that c is the minimum of the two values. u(t) =


Extracted text: where c = and d 1 c) Use a graphing utility to plot the solution for k , k = 1, and 2' k = 2. Describe the principal features of the solution and how they depend on k. In all cases the plot will be zero for 0 < ·="" for="" くtく="" the="" plot="" will="" be="" the="" system="" 2="" response="" to="" a="" step="" input="" of="" magnitude="" k.="" when="" k="" increases="" the="" amplitude="" increases.="" d)="" use="" a="" graphing="" utility="" to="" find,="" to="" two="" decimal="" places,="" the="" smallest="" value="" of="" k="" for="" which="" the="" solution="" u(t)="" reaches="" the="" value="" 1.="" k="2" e)="" suppose="" k="2." use="" a="" graphing="" utility="" to="" find="" the="" time="" t,="" to="" two="" decimal="" places,="" after="" which="" |u(t)|="">< 0.1="" for="" all="" t=""> T. 312