Question

The differential equation representing the family of parabolas having vertex at the origin and axis along the positive Y-axis is

• A. $x\frac{dy}{dx}-2y=0$
• B. $\frac{dy}{dx}+xy=0$
• C. $x\frac{dy}{dx}+y=0$
• D. $x^{2}\frac{dy}{dx}+y=0$

Hint:

Eliminate the arbitrary constant 'a' to find the differential equation.

Answer 1

The Correct Option is A

Step 1: Concept
The equation of such a parabola is $x^{2} = 4ay$.

Step 2: Analysis
Differentiate with respect to $x$: $2x = 4a\frac{dy}{dx}$.
From the original equation, $4a = x^{2}/y$.

Step 3: Calculation
Substitute $4a$: $2x = \frac{x^{2}}{y}\frac{dy}{dx}$
$2y = x\frac{dy}{dx} \implies x\frac{dy}{dx} - 2y = 0$.

Step 4: Conclusion
Hence, the correct differential equation is $x\frac{dy}{dx}-2y=0$. Final Answer: (A)