Consider the multivariate function $f(x,y)=ln(x^2+y^2).$

$($a$)(10$ points$)$ Compute the four second$-$order partial derivatives.

$($b$)(5$ points$)$ Verify that the mixed partial derivatives are equal.

$($c$)(5$ points$)$ Determine whether Laplace's equation holds; that is$,$ whether

$\frac{de{l}^{2}f}{del{x}^2}+\frac{de{l}^{2}f}{del{y}^2}=0$