C. $ y(x) = C_1 \sin(2x) + C_2 \cos(2x) $

["SEO-Optimized Article: Understanding ( y(x) = C_1 \sin(2x) + C_2 \cos(2x) ) – The General Solution to a Second-Order Differential Equation", "---", "### Introduction\nThe function ( y(x) = C_1 \sin(2x) + C_2 \cos(2x) ) is a fundamental solution to a second-order linear homogeneous differential equation with constant coefficients. Widely used in engineering, physics, and applied mathematics, this expression plays a crucial role in modeling oscillatory systems such as mechanical vibrations, electromagnetic waves, and signal processing. In this SEO-optimized article, we explore the mathematical meaning, derivation, properties, and applications of this classic trigonometric solution.", "---", "### What is ( y(x) = C_1 \sin(2x) + C_2 \cos(2x) )?", "The equation describes a simple harmonic motion where:\n- ( C_1 ) and ( C_2 ) are arbitrary constants determined by initial conditions,\n- ( \sin(2x) ) and ( \cos(2x) ) are periodic basis functions with angular frequency ( \omega = 2 ),\n- The solution represents the general form of the system’s response under undamped, sinusoidal forcing.", "---", "### The Differential Equation Behind the Solution", "This solution arises from solving the differential equation:\n[\ny'' + 4y = 0\n]\nwhere ( y'' = \frac{d^2y}{dx^2} ).", "Why frequency 2?\nThe coefficient 4 in ( y'' + 4y = 0 ) comes from squaring the angular frequency ( \omega = 2 ) (since characteristic equation ( r^2 + \omega^2 = 0 \implies r = \pm i\omega )). This leads naturally to sine and cosine terms.", "---", "### Key Mathematical Properties", "1. Linear Combination of Basis Functions\nThe general solution shows ( y(x) ) is a linear combination of ( \sin(2x) ) and ( \cos(2x) ), forming a 2-dimensional solution space. Any solution to this equation can be expressed this way, hence the term “general solution.”", "2. Periodicity\nWith ( \omega = 2 ), the function has period:\n[\nT = \frac{2\pi}{2} = \pi\n]\nThis periodic nature makes it ideal for modeling repeating waveforms.", "3. Amplitude-Phase Representation\nUsing trigonometric identities, ( y(x) ) can be rewritten as:\n[\ny(x) = R \sin(2x + \phi)\n]\nwhere ( R = \sqrt{C_1^2 + C_2^2} ), and ( \phi = \ an^{-1}\left(\frac{C_2}{C_1}\right) ). This reveals an amplitude ( R ) and phase shift ( \phi ), simplifying interpretation.", "---", "### How to Use This Solution in Practice", "- Initial Conditions: Use ( y(0) ) and ( y'(0) ) to solve for ( C_1 ) and ( C_2 ).\n- Engineering Applications:\n - Model simple harmonic oscillators with no damping\n - Analyze two-coupled vibration systems\n - Design control systems using sinusoidal steady-state responses\n- Signal Processing: Common in Fourier series and wave decomposition.", "---", "### Visualizing the Function", "Imagine plotting ( y(x) ) vs. ( x ): smooth, undulating waves repeated every ( \pi ) units. The shape depends on the constants ( C_1 ) and ( C_2 ): for example,\n- ( C_1 = R, C_2 = 0 \Rightarrow y(x) = R \sin(2x) )\n- ( C_1 = 0, C_2 = R \Rightarrow y(x) = R \cos(2x) )\n- A mix gives a shifted, amplitude-modulated wave.", "---", "### Why This Solution Is Important in STEM", "- Physics: Models wave propagation, LC circuits, and pendulum motion near equilibrium.\n- Engineering: Essential in vibration analysis and structural dynamics.\n- Mathematics: Exemplifies solutions to linear homogeneous ODEs with complex roots in the characteristic equation.\n- Computer Science: Used in algorithms for signal filtering and frequency analysis.", "---", "### Frequently Asked Questions", "Q: Why isn’t the solution just sine or cosine?\nA: The general solution requires two linearly independent solutions. Since ( \sin(2x) ) and ( \cos(2x) ) are independent, their linear combination accounts for all possible behaviors of the system.", "Q: Can ( C_1 ) and ( C_2 ) be any real numbers?\nA: Yes, within the context of the problem, ( C_1, C_2 \in \mathbb{R} ) fully determine the solution’s amplitude and phase.", "Q: Is this solution periodic?\nA: Yes, with period ( \pi ), due to the angular frequency of 2.", "---", "### Conclusion", "The expression ( y(x) = C_1 \sin(2x) + C_2 \cos(2x) ) is more than just a math identity—it’s a powerful, versatile tool that bridges theory and real-world applications. From understanding mechanical oscillations to designing electronic circuits, mastering this solution enables students, engineers, and scientists to analyze and predict dynamic systems with clarity and precision.", "---", "### Related SEO Keywords\n- Second-order differential equation solution\n- Sinusoidal function ( y(x) )\n- General solution harmonic motion\n- Sinusoidal steering in engineering\n- Two-dimensional ODE solutions\n- Phase and amplitude in harmonic motion\n- Undamped harmonic oscillator model", "---", "Meta Description:\nDiscover why ( y(x) = C_1 \sin(2x) + C_2 \cos(2x) ) is the standard general solution to ( y'' + 4y = 0 ), its mathematical properties, and applications in physics and engineering. Learn how this trigonometric form enables analysis of oscillatory systems and wave phenomena.", "---", "Keywords: ( y(x) = C_1 \sin(2x) + C_2 \cos(2x) ), differential equation solution, harmonic motion, oscillation, amplitude-phase form, physics applications, engineering dynamics, linear ODEs."]