Serway & Jewett — Wave Motion: Problem 22

(a) Show that the function \( y(x, t) = x^2 + v^2 t^2 \) is a solution to the wave equation. (b) Show that the function in part (a) can be written as \( f(x + vt) + g(x - vt) \) and determine the functional forms for \( f \) and \( g \). (c) What If? Repeat parts (a) and (b) for the function \( y(x, t) = \sin (x) \cos (vt) \).