Question

If \(y=c_{1}\sin ^{2}x+c_{2}\), then its differential equation is:

• A. \(\frac{d^{2}y}{dx^{2}}+2\,\cot 2x\frac{dy}{dx}=0\)
• B. \(\frac{d^{2}y}{dx^{2}}-\cot 2x\frac{dy}{dx}=0\)
• C. \(\frac{d^{2}y}{dx^{2}}-2\,\cot x\frac{dy}{dx}=0\)
• D. \(\frac{d^{2}y}{dx^{2}}-2\,\cot 2x\frac{dy}{dx}=0\)

Hint:

Differentiate twice and eliminate the constant c1 by dividing the second derivative by the first.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
There are two arbitrary constants, so differentiate twice and eliminate them.

Step 2: First derivative
\(\frac{dy}{dx}=c_{1}\cdot2\sin x\cos x=c_{1}\sin2x\).

Step 3: Second derivative
\(\frac{d^{2}y}{dx^{2}}=2c_{1}\cos2x\).

Step 4: Eliminate c1
Dividing: \(\frac{y''}{y'}=2\cot2x\), so \(\frac{d^{2}y}{dx^{2}}-2\cot2x\frac{dy}{dx}=0\).

Final Answer:
The differential equation is y'' − 2 cot 2x · y' = 0. \[ \boxed{\text{(D) }\dfrac{d^{2}y}{dx^{2}}-2\cot 2x\dfrac{dy}{dx}=0} \]