What is the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis ?
- A. x\frac{dy}{dx}+2y=0
- B. x\frac{dy}{dx}-2y=0 ✓
- C. y\frac{dx}{dy}+2x=0
- D. y\frac{dx}{dy}-2x=0
Correct Answer: B. x\frac{dy}{dx}-2y=0
Explanation
The equation of such a family of parabolas is x^2 = 4ay. Differentiating with respect to x gives 2x = 4a \frac{dy}{dx}. Substituting 4a = \frac{x^2}{y} into the differentiated equation yields 2x = \frac{x^2}{y} \frac{dy}{dx}. Dividing by x and rearranging gives x\frac{dy}{dx} - 2y = 0.
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