 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-2/12/b/solution

ω = k / m

x = A cos ω t + B sin ω t, y = C cos ω t + D sin ω t .

The four initial data give

A = X_{0}

B = C = 0

, and

D = V_{0} / ω

. Thus

x (t) = X_{0} cos ω t, y (t) = \frac{V _{0}}{ω} sin ω t .

This is the position component of the isotropic harmonic oscillator flow. Substitution also gives the constant energy

E = (k X_{0}^{2} + m V_{0}^{2}) /2

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