$($b$)$ Let $y=f(xat)(x-at)$ where $a>0$ is constant, fand $$ are two functions with continuous derivatives. Show thaty is a solution to the partial equation.

$\frac{de{l}^{2}y}{del{t}^2}=a^2\frac{de{l}^{2}y}{del{x}^2}$

$($c$)$ Suppose that $y$ is a function of one variable $x$ determined by the following equation

$ln\sqrt[\phantom{2}]{x^2{y}^2}=arctan(\frac{y}{x})$

Find The derivatives $dy$ and $a^{2}y$

$dx,d{x}^2$

in terms of $x$ and $y.$