# Since i 0 0 then c v r the solution to the ivp is

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Since I (0) = 0, then C = - V R . The solution to the IVP is therefore I ( t ) = V R 1 - e - Rt L . The final value is obtained as t approaches infinity. Therefore, the final value is V R . We are looking for t ? such that I ( t ? ) = 0 . 95 V R . This gives t ? = L R ln(20) .
3: Mass-Spring system Consider a mass-spring system. The displacement of the mass from equilibrium is modelled by a function x ( t ). Assuming that the mass is 1 unit, the spring constant is k > 0, the damping coefficient is 2 d where d > 0, then x ( t ) satisfies the ODE: ¨ x + 2 d ˙ x + kx = 0 , 2
ECE 205 - Winter 2017 Assignment 3 Solutions Due 05-02-2017 where ˙ x is the first derivative of the position with respect to time and ¨ x is the second derivative of the position with respect to time. (a) What choices for d yield underdamped oscillations? In this regime, what is the angular frequency of oscillations?
(b) If the value of d is chosen such that the system is on the boundary of being underdamped then we say that the system is critically damped. Suppose that the system is critically damped, and we have the initial values x (0) = 1 and ˙ x (0) = v . Find the particular solution solving this IVP in terms of v . Show that no matter what the initial velocity v is, the mass can never pass the equilibrium point more than once.
4: Almost Simple Harmonic Oscillator Consider the ODE y 00 ( t ) + 2 λy 0 ( t ) + ω 2 0 y ( t ) = 0 . How many oscillations take place until the amplitude decays by 1% when: