Final answer:
The general solution of oscillatory differential equations often includes a combination of sine and cosine functions. Differential equations that describe wave phenomena may use such solutions to express factors like displacement, velocity, and acceleration.
Step-by-step explanation:
To find the general solution of a differential equation such as (2x sin y cos y)y = 5x² + sin² y, we need to employ techniques of differential equations, which may include separation of variables, integrating factors or potentially other methods depending on the form of the equation. However, without more context or the exact form of the equation, it is difficult to provide a specific solution method. When encountering oscillatory functions in the context of wave equations, the general solution often involves a combination of sines and cosines as indicated by Yk (x) = Ak cos kx + Bk sin kx, where k is the wave number and Ak, Bk are the amplitude coefficients for the cosine and sine terms, respectively.
In the case of wave functions such as y (x, t) = A sin (kx - wt), their sums can also be solutions to linear wave equations, underlining the principle of superposition. Moreover, properties like the average value of sin² over a cycle equaling that of cos² also become relevant in analyzing these functions.
When specific values are given for wave equations, one can determine various characteristics of the wave such as velocity, displacement, and acceleration at particular instances and positions.