Final answer:
The general solution to the given differential equation y''+4y=0 is an oscillatory function, reflective of sinusoidal wave forms such as sines and cosines. Such solutions are essential in understanding harmonic motion and wave phenomena in physics.
Step-by-step explanation:
The student's question pertains to solving the differential equation y''+4y=0. This is a second-order linear homogeneous differential equation with constant coefficients, common in the study of harmonic motion and wave equations.
A general solution to such an equation is indeed oscillatory, typically in the form of sines and cosines due to their periodic nature.
For instance, the general solution provided in the reference information, Yk (x) = Ak cos kx + Bk sin kx, represents combinations of sine and cosine functions that solve the differential equation where Ak and Bk are arbitrary constants and k is a constant that relates to the frequency of oscillation.
Examples from wave physics illustrate how wave functions, such as A sin (kx - wt) and y1 (x, t) + y2 (x, t) = A sin (kx − wt + φ) with different phases, can combine to form solutions to linear wave equations.
Superposition of waves and the resultant wave velocity can also be described by these solutions, influencing wave properties such as amplitude and phase.