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(a) Find the general solution of the motion of a mass attached to the ceiling by a spring in presence of friction, i.e. solve the ODE mimg-k(y-l) - yy. with m = 1, k = 3, y = 2, g = 10, 1 = 5, where y indicates the distance of the mass from the ceiling. (b) What is the limit limt→y(t) for the motion of the mass described in (a)? Describe in words the asymptotic dynamical behaviour of the mass for t→∞. (c) Determine whether the differential equation 1 29² + y cos(x) + (yx + sin(x) - e") y = 0 is exact. If it is exact, find its general solution in explicit form. [3

(a) Find the general solution of the motion of a mass attached to the ceiling by a spring in presence of friction, i.e. solve the ODE mimg-k(y-l) - yy. with m = 1, k = 3, y = 2, g = 10, 1 = 5, where y indicates the distance of the mass from the ceiling. (b) What is the limit limt→y(t) for the motion of the mass described in (a)? Describe in words the asymptotic dynamical behaviour of the mass for t→∞. (c) Determine whether the differential equation 1 29² + y cos(x) + (yx + sin(x) - e") y = 0 is exact. If it is exact, find its general solution in explicit form. [3

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) Find the general solution of the motion of a mass attached to the ceiling by a spring in presence of friction, i.e. solve the ODE mi = mg -k(y-l) — yy. with m = 1, k = 3, y = 2, g = 10, 1 = 5, where y indicates the distance of the mass from the ceiling. (b) What is the limit lim y(t) for the motion of the mass described in (a)? Describe in words the asymptotic dynamical behaviour of the mass for t→∞. (c) Determine whether the differential equation 8x 12 y ² + y cos(x) + (yx + sin(x) — e³) y' = 0 - is exact. If it is exact, find its general solution in explicit form. [1
Transcribed Image Text:(a) Find the general solution of the motion of a mass attached to the ceiling by a spring in presence of friction, i.e. solve the ODE mi = mg -k(y-l) — yy. with m = 1, k = 3, y = 2, g = 10, 1 = 5, where y indicates the distance of the mass from the ceiling. (b) What is the limit lim y(t) for the motion of the mass described in (a)? Describe in words the asymptotic dynamical behaviour of the mass for t→∞. (c) Determine whether the differential equation 8x 12 y ² + y cos(x) + (yx + sin(x) — e³) y' = 0 - is exact. If it is exact, find its general solution in explicit form. [1
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