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Consider the initial value problem mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0 80 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t) where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k Newtons per meter, and F(t) = 80 cos(8t) Newtons. Solve the initial value problem. -4t 3 x(t) = e 12 cos (81) - ½ cos(81)) - cos (8t) + sin (81) help (formulas) Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0? If it is, enter zero 0047 If not, enter a function that approximates x(t) for very large positive values of t. For very large positive values of t, x(t) sp(t)=0 help (formulas) ≈ x

Consider the initial value problem mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0 80 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t) where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k Newtons per meter, and F(t) = 80 cos(8t) Newtons. Solve the initial value problem. -4t 3 x(t) = e 12 cos (81) - ½ cos(81)) - cos (8t) + sin (81) help (formulas) Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0? If it is, enter zero 0047 If not, enter a function that approximates x(t) for very large positive values of t. For very large positive values of t, x(t) sp(t)=0 help (formulas) ≈ x

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 59E: According to the solution in Exercise 58 of the differential equation for Newtons law of cooling,...
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Consider the initial value problem mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and F(t) = 80 cos(8t) Newtons. Solve the initial value problem. 3 x(t) = e(cos(81) - ½ cos(81)) - cos(8) + sin(8) help (formulas) Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0? If it is, enter zero too If not, enter a function that approximates x(t) for very large positive values of t. For very large positive values of t, x(t)=xsp(t) = 0 help (formulas)
Transcribed Image Text:Consider the initial value problem mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and F(t) = 80 cos(8t) Newtons. Solve the initial value problem. 3 x(t) = e(cos(81) - ½ cos(81)) - cos(8) + sin(8) help (formulas) Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0? If it is, enter zero too If not, enter a function that approximates x(t) for very large positive values of t. For very large positive values of t, x(t)=xsp(t) = 0 help (formulas)
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