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u" + 2u' + 10u = kg(t), u(0) = 0, u'(0) = 0, where g(t) = us(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 5) Solve the initial value problem. VOTE: Denote the two Heaviside functions by uc(t) and ua(t). Indicate separately he values of c and d knowing that c is the minimum of the two values. (t):

u" + 2u' + 10u = kg(t), u(0) = 0, u'(0) = 0, where g(t) = us(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 5) Solve the initial value problem. VOTE: Denote the two Heaviside functions by uc(t) and ua(t). Indicate separately he values of c and d knowing that c is the minimum of the two values. (t):

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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) =
Transcribed Image 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) =
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 <t < · 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
Transcribed Image 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 <t < · 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
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