Show that `(a)y=(x+vt)^(2),(b)y=(x+t)^(2),(c )y=(x-vt)^(2)`, and (d) y=2 sin xcos vt are each a solution of one dimensional wave equation but not (e) y=x^(2)-v^(2)t^(2)` and (f) `y=sin 2x cos vt.`

Explanation

To show that the given functions are solutions of the one-dimensional wave equation, we will use the wave equation in the form:

\[ \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2} \]

where \(y\) is a function of \(x\) and \(t\), and \(v\) is the wave speed.

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