Final answer:
The order of the differential equation dx/dy − xy^2 = 0 is first, and it is nonlinear because it contains a product of the variable y and its derivative, which cannot be expressed in a standard linear form.
Step-by-step explanation:
The differential equation given in the student's question is dx/dy − xy^2 = 0. To determine its order, we look for the highest derivative in the equation, which in this case is dx/dy. As this is a first derivative, the order of the equation is first. To determine if the equation is linear or nonlinear, we assess whether it can be written in the standard linear form, which is dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions or constants. As the term −xy^2 cannot be expressed in this form since it involves a product of the dependent variable y and its derivative, the differential equation is nonlinear. Therefore, the correct answer is (a) 1st order; nonlinear.