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Maps - + UD AN D V 2ow) RR = x= ==[(en + 03" 2. (60%) Consider the following ODE which models the vibration of a gas bubble in liquid: 3 RO + - R2 20w 4uR Po 2 RO Po - P(t) R R where p is the density of the surrounding liquid, po is atmospheric pressure (1.01 - 105 Pa), ow is the surface tension at an air-water interface (0.072 N/m), y is a constant equal to 1.0, My is the viscosity of the surrounding liquid, P(t) is an externally applied time-varying pressure wave, and the dots denote differentiation with time. Write a Python program that solves this ODE from 0 <t < 50.10+ s (use dt = 10-8 s) for a gas bubble of initial radius Ro in water, where p = 1000 kgm?, M = 0.001 Pa·s, and is driven by a pressure wave of P(t) = 5x103sin (2nft) where f = 106 Hz. Assume that the initial velocity Ř(0) is 0. For full marks, have your program: a) Using a loop, solve this equation three times, once for each bubble of a given initial size of Ro = 2.7.10-6 m, Ro = 3.1.10-6m and Ro = 3.5. 10-6 m. Plot and overlay the results from all three t - 1 and the x-axis is time from 25 <t< 50 us. Label the axes, include a legend, and give the title of the plot your last name. b) In a second figure, plot the magnitude of the Fourier transform of the data presented in part a (the solutions for all three bubbles from 25 st s 50 us on a single graph). Set the x-axis limits from 0.5 MHz to 3.5 MHz. Include a legend and appropriate x and y axis labels. c) If we call the magnitude of the Fourier transform at f = 106 Hz as Ay, and the magnitude of the A Fourier transform at f = 2 · 106 Hz as Az, print out the ratio 42/A, for Ro = 3.5 um using f- string notation to 3 decimal places. of these simulations, where the y-axis is (R(t)/Ro) - ENG US ASUS ZenBook