Final answer:
The differential equation of the family of parabolas with focus at the origin and the x–axis as axis is y(dy/dx)²+2xy(dy/dx)+y=0. This equation is derived by differentiating the equation of a parabola with focus at the origin and x-axis as the axis of symmetry, and substituting the derivatives into the given options. Option D is the only option that satisfies the differential equation.
Step-by-step explanation:
The differential equation of the family of parabolas with focus at the origin and the x–axis as axis is y(dy/dx)² + 2xy(dy/dx) + y = 0 (Option D).
To understand why this is the correct answer, we can consider the equation of a parabola with the focus at the origin and x-axis as the axis of symmetry. Such an equation takes the form y = ax + bx². When we differentiate this equation with respect to x, we get dy/dx = a + 2bx. Substituting these values into the differential equation options, we can see that only Option D satisfies the equation.