Which of the following is the differential equation for the family of parabolas with their vertex at the origin and focus on the positive x-axis?

Which of the following is the differential equation for the family of parabolas with their vertex at the origin and focus on the positive x-axis?

• A. y² = 2xy'
• B. x² = 2yy'
• C. y = xy'
• D. x = 2yy'

Answer: A) y² = 2xy'

Explanation: The family is y² = 4ax. Differentiating w.r.t x: 2y(dy/dx) = 4a. Substitute 4a = (2y)dy/dx into the original equation: y² = x(2y)dy/dx → y² = 2xy(dy/dx).