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For all real x, let y(x) be the solution of the differential equation dy/dx + 12y = cos((pi/12) x) with y(0) = 0. Which one of the following statements about y(x) is correct?

1. There is a real value beta for which the horizontal line y = beta meets the curve y = y(x) at infinitely many points
2. y(x) is an increasing function for all x
3. y(x) is a decreasing function for all x
4. y(x) is a periodic function

Correct answer: There is a real value beta for which the horizontal line y = beta meets the curve y = y(x) at infinitely many points

Solution

The solution consists of an exponentially decaying term plus a steady sinusoidal oscillation of period 24. Because of the oscillation it is neither monotonic (rules out increasing/decreasing) nor exactly periodic (the decaying term breaks periodicity), but its long-run oscillation crosses a suitable horizontal line y = beta infinitely many times.

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