$($a$)$ Give state diagrams of DFA$$s recognizing the following languages. The alphabet is $\{$x$,$ y$\}.$ Please make sure to draw the diagrams!

$$ L$1=\{$w$|$ w has length $4$ and its second symbol is a x$\}$

$$ L$2=\{$w$|$ w ends with x$\}$

$$ L$3=\{$w$|$ w the length of w is at most $4\}$

$$ L$4=\{$w$|$ has at least $2$ x$$s and at most $2$ y$$s$\}$

$($b$)$ Write the formal definition of L$2$