# Numerical solutions to Different equations

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Science

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Numerical solutions to Diff. Eqs.

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JRT 1.50 (modified). Numerical solutions to Diff. Eqs. The differential equation for the oscillating skateboard (JRT Ex. 1.2) given by Eq. (1.51) g φ̈ = − sin φ R cannot be solved in terms of elementary functions. The period of oscillation can be written in terms of elliptical integrals and their numerical solutions, e.g., Jacobi elliptic functions. Modern computational programs afford easier solutions to the pendulum oscillator problem for starting amplitudes not necessarily small. Note: this example problem shares an equation of motion with a simple pendulum. (a)Using Mathematica (or similar symbolic/numeric mathematical software), solve the differential equation above for the case of releasing the skateboard from rest at a starting angle φ0 = π/4 and using the same values for R and g used in Ex. 1.2, specifically R = 5 m and g = 9.8 m/s2. (b)Plot φ versus time t for at least three full periods. Be sure to label your axes, including units and numerical values. Additionall add a curve showing the approximate solution for the oscillator problem with the same initial conditions; be sure to label each curve clearly. Comment on similarities and differences of the two, especially in light of the fact that φ0 = π/4 is not especially small compared to 1. . Hint: Use NDSolve in Mathematica to numerically solve the differential equation. The extensive help included inMathematica provides information on syntax, examples, etc. Simply type in the notebook ? NDSolve followed by SHIFT-ENTER to launch the help specifically for NDSolve. Click on the
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Attached.

ϕi=ϕi-1+ 1/2(wi+wi-1)(ti-ti-1)
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