Discussion
The general solution to the ordinary differential equation \(\frac{d^{2}y}{dx^{2}} = \cos 2x\) is
Answer: A
The general solution is \(y = -\frac{\cos 2x}{4} + C_{1}x + C_{2}\).
Integrate twice:
\[ \begin{aligned} \frac{dy}{dx} &= \int \cos 2x\, dx = \frac{\sin 2x}{2} + C_{1} \\ y &= \int \left( \frac{\sin 2x}{2} + C_{1} \right) dx \\ &= -\frac{\cos 2x}{4} + C_{1}x + C_{2} \end{aligned} \]
A second-order equation needs two arbitrary constants, which rules out both options with only \(C_{1}\); \(\frac{\sin 2x}{2} + C_{1}\) is in fact only the first integral. The option with \(+\frac{\cos 2x}{4}\) has the wrong sign: its second derivative is \(-\cos 2x\).
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