Verify that the piecewise$-$defined function

y $=$

$6$x$2,$ x $<06$x$2,$ x $0$

is a solution of the differential equation

xy$2$y $=0$

on $($$,$ $).$

Begin by finding the piecewise$-$defined function

y$.$

y$=$

$,$ x $<0$

$,$ x $0$

For

x $<0,$

in terms of x$,$

xy$2$y$=$x $$

$2(6$x$2)=$

$+12$x$2=$

$.$

For

x $0,$

in terms of x$,$

xy$2$y$=$x $$

$2(6$x$2)=$

$12$x$2=$

$.$