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A mass of 30 kg swings from a pendulum with length 25 meters. Assume that the acceleration due to gravity is 9.8 meters per sec per sec. Let 0(t) be the angle swept by the pendulum after t seconds and find a differential equation for ''(t). 0'' (t) = We will obtain approximate solutions to this nonlinear ODE in two different ways. a) First use the linear approximation for small values given by the Maclaurin polynomial sin(0) 0 to write an approximate linear ODE, which you should solve analytically subject to the initial conditions (0) = = 0.2 and 0' (0) = 2. Then evaluate at 0.3 seconds, which gives your first approximation for the angle swept by the pendulum after 0.3 seconds. 0(0.3)~ b) Second, return to the original nonlinear ODE and use the integrating factor 20' to transform the 2nd order ODE into a 1st order ODE. Express this equation in the form 0' (t) = √a + bcos (0) to the initial conditions (0) = 0.2 and 0'(0) = - 2. Round the coefficients in your final answer to six decimal places. 0' (t) c) The nonlinear equation from part (b) is not solvable with elementary functions, but you can approximate its solution. Using a step size of 0.06 and the Improved Euler (Heun's) Method, approximate the angle swept by the pendulum after 0.3 seconds using the model from part (b). Make sure to use your rounded coefficients and the initial conditions (0) - 0.2 and 0'(0) = -2. = 0(0.3)~