Question

The differential equation whose solution represents the family $x^2y = 4e^x + c$, where c is an arbitrary constant, is

• A. $x \frac{dy}{dx} + xy = 0$
• B. $x^2 \frac{dy}{dx} + (2x - xy) = 0$
• C. $x \frac{dy}{dx} + (x - 2)y = 0$
• D. $x^2 \frac{dy}{dx} + 2xy - 4e^x = 0$

Hint:

If a constant "c" is additive, it disappears upon the first differentiation. No substitution is needed.

Answer 1

The Correct Option is D

Step 1: Concept
Differentiate the given equation with respect to $x$ to eliminate the arbitrary constant $c$.

Step 2: Meaning
Use the product rule on $x^2y$: $\frac{d}{dx}(x^2y) = x^2 \frac{dy}{dx} + y(2x)$.

Step 3: Analysis
$\frac{d}{dx}(x^2y) = \frac{d}{dx}(4e^x + c)$. $x^2 \frac{dy}{dx} + 2xy = 4e^x + 0$. Rearranging: $x^2 \frac{dy}{dx} + 2xy - 4e^x = 0$.

Step 4: Conclusion
The resulting differential equation is $x^2 \frac{dy}{dx} + 2xy - 4e^x = 0$. Final Answer: (D)