Determine whether y = xex is a solution to the d.e.
y' = xy.
If y = xex then
y' = xex + ex.
This is the left-hand side of the differential equation. The right-hand side of the differential equation is
xy = xex.
Since the left- and right- hand sides of the d.e. are not equal when y = xex, this is NOT a solution to the differential equation.