Discussion
The equation of motion for the forced simple harmonic oscillator is
\(\ddot{x}(t) + \omega^{2}x(t) = F\cos(\omega t)\)
where \(x(t = 0) = 0\) and \(\dot{x}(t = 0) = 0\). Which one of the following options is correct?
Answer: A
The solution grows as x(t) ∝ t sin(ωt).
The driving frequency equals the natural frequency, so the system is at resonance. Try x = Ct sin ωt:
\[ \begin{aligned} \ddot{x} &= 2C\omega\cos\omega t - C\omega^{2}t\sin\omega t \\ \ddot{x} + \omega^{2}x &= 2C\omega\cos\omega t \Rightarrow C = \frac{F}{2\omega} \end{aligned} \]
This particular solution also satisfies x(0) = 0 and ẋ(0) = 0, so no homogeneous part is needed: x(t) = (F/2ω) t sin ωt, an oscillation whose amplitude grows linearly in time.
t cos ωt does not satisfy ẋ(0) = 0 and does not solve the equation. The amplitude grows without bound but x(t) is finite at every time, so 'x(t) = ∞' is wrong, and there is no exponential growth for an undamped linear oscillator.
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