The problem y'=cos(xy) is NOT SEPARABLE. There are two ways to check this. 1. Compute the partial of f(x,y) on y, divided by f(x,y). This equals -x sin(xy) / cos(xy) = -x tan(xy). At y=1 it becomes -x tan(x), so it depends on x. Then apply the test: NOT SEPARABLE. The blackboard chalk on this one has f_x written near the end, which should be f_y like the previous line. Notation: f_x means partial of f on x, f_y means partial of f on y. 2. Compute the partial of f(x,y) on x, divided by f(x,y). This equals -ysin(xy) / cos(xy) = -y tan(xy). At x=1 it becomes - tan(y), so it depends on y. Then apply the test: NOT SEPARABLE. Only one of these tests is needed to conclude the equation is not separable. The two possibilities are shown, because both solutions are valid. In this problem, details were not requested. So none need to be displayed. The answer: LEAVE THE BOX BLANK.