Analysis of Solution to First Order Linear Differential Equation

To analyze the given first-order linear differential equation $\frac{dy}{dx} + 12y = \cos\left(\frac{\pi}{12} x\right), \quad y(0) = 0$

we compute the integrating factor $I(x) = e^{\int 12 \, dx} = e^{12x}$. Solving the differential equation yields the explicit solution: $y(x) = \frac{1}{144 + \frac{\pi^2}{144}} \left(12 \cos\left(\frac{\pi}{12} x\right) + \frac{\pi}{12} \sin\left(\frac{\pi}{12} x\right) - 12 e^{-12x}\right)$

Analyzing the properties of $y(x)$:

1. Monotonicity (Options A and B): The solution contains sinusoidal terms that cause $y(x)$ to oscillate infinitely as $x \to \infty$, making $y(x)$ neither strictly increasing nor strictly decreasing on $\mathbb{R}$.
2. Periodicity (Option D): The presence of the transient decaying exponential term $-12 e^{-12x}$ breaks the periodicity of the function, so $y(x)$ is not periodic.
3. Intersections with $y = \beta$ (Option C): For $\beta = 0$, the equation $y(x) = 0$ simplifies to an equation where the oscillating sinusoidal component balances the exponentially decaying term, yielding infinitely many real solutions as $x \to \infty$.

Thus, statement (C) is the correct choice.

Correct Answer: (C)