Question

The differential equation whose solution represents the family \(x^2 y=4 \mathrm{e}^x+\mathrm{c}\), where c is an arbitrary constant, is

• A. \(x\frac{\text{ }\mathrm{d}y}{\text{ }\mathrm{d}x}+xy=0\)
• B. \(x^{2}\frac{\text{ }\mathrm{d}y}{\text{ }\mathrm{d}x}+(2x-xy)=0\)
• C. \(x\frac{\text{ }\mathrm{d}y}{\text{ }\mathrm{d}x}+(x-2)y=0\)
• D. \(x^{2}\frac{\text{ }\mathrm{d}y}{\text{ }\mathrm{d}x}+2xy-4\mathrm{e}^{x}=0\)

Hint:

Differentiate once to eliminate the arbitrary constant c.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
One arbitrary constant means the differential equation is of first order. Differentiate the family once and remove c.

Step 2: Differentiate
From \(x^{2}y=4e^{x}+c\), differentiate with respect to \(x\): \(x^{2}\dfrac{dy}{dx}+2xy=4e^{x}\).

Step 3: Constant removed
The constant c vanishes on differentiation, so no further elimination is needed.

Step 4: Rearrange
\(x^{2}\dfrac{dy}{dx}+2xy-4e^{x}=0\).

Final Answer:
The differential equation is x^2 dy/dx + 2xy - 4e^x = 0. \[ \boxed{\text{(D) }x^{2}\dfrac{dy}{dx}+2xy-4e^{x}=0} \]