47. Find the cubic spline interpolation at x = 1.5 based on the data set x = = [0, 1, 2], y = [1, 3, 2]. 53. Consider a pendulum of length 1 with a mass, m, at its end (see the figure from the note). The angle the pendulum makes with the vertical axis over time, (t), in the presence of vertical gravity, g, can be described by the pendulum equation, which is the ODE ml d²(t) dt² = ―mg sin((t)). If we assume the angles are very small (i.e., sin(③(t)) ≈ ✪(t)), then the pendulum equation reduces to dᎾ(t) dt² = -ge(t). (a) Find a general solution to the pendulum equation. (b) If the angle and angular velocities de(t) dt at t = 0 are the known values, O。 and 0, respectively, find a particular solution for these known values. (c) Reduce the second order pendulum equation to first order, where [e(t)] S(+) = [80] (d) Write the ODE in matrix form.

47. Find the cubic spline interpolation at x = 1.5 based on the data set x = = [0, 1, 2], y = [1, 3, 2]. 53. Consider a pendulum of length 1 with a mass, m, at its end (see the figure from the note). The angle the pendulum makes with the vertical axis over time, (t), in the presence of vertical gravity, g, can be described by the pendulum equation, which is the ODE ml d²(t) dt² = ―mg sin((t)). If we assume the angles are very small (i.e., sin(③(t)) ≈ ✪(t)), then the pendulum equation reduces to dᎾ(t) dt² = -ge(t). (a) Find a general solution to the pendulum equation. (b) If the angle and angular velocities de(t) dt at t = 0 are the known values, O。 and 0, respectively, find a particular solution for these known values. (c) Reduce the second order pendulum equation to first order, where [e(t)] S(+) = [80] (d) Write the ODE in matrix form.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 94E

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