Final answer:
The linear combination x(t) = x₁(t) x₂(t) is a solution of the simple harmonic oscillator equation due to the principle of superposition.
Step-by-step explanation:
According to the principle of superposition, if the wave functions y₁ (x, t) = f (x = ut) and y₂ (x, t) = g(x = vt) are solutions to the linear wave equation, then the linear combination Ay₁ (x, t) + By₂ (x, t), where A and B are constants, is also a solution to the linear wave equation. This means that if x(t) = x₁(t) x₂(t) is a linear combination of two wave functions that are solutions to the simple harmonic oscillator equation, then x(t) is also a solution to the simple harmonic oscillator equation.