The differential equation d2xdy2+y+cot2⁡x=0 must be satisfied by

2 + c 1 cos ⁡ x + c 2 sin ⁡ x + cos ⁡ x log ⁡ tan ⁡ x 2 , all the above

The differential equation d2xdy2+y+cot2⁡x=0 must be satisfied by

( a ) : y = 2 + c 1 cos ⁡ x + c 2 sin ⁡ x d y d x = − c 1 sin ⁡ x + c 2 cos ⁡ x d 2 y d x 2 = − c 1 cos ⁡ x − c 2 sin ⁡ x = 2 − y d 2 y d x 2 + y + 2 = 0 ( b ) : y = cos ⁡ x ln ⁡ tan ⁡ x 2 d y d x = cos ⁡ x sec 2 ⁡ ( x / 2 ) 2 tan ⁡ ( x / 2 ) − sin ⁡ x ln ⁡ tan ⁡ x 2 d y d x = cot ⁡ x − sin ⁡ x ln ⁡ tan ⁡ x 2 d 2 y d x 2 = − cot 2 ⁡ x − 2 − cos ⁡ x ln ⁡ tan ⁡ x 2 d 2 y d x 2 + y + cot 2 ⁡ x = 0 ( c ) : y = 2 + c 1 cos ⁡ x + c 2 sin ⁡ x + cos ⁡ x ln ⁡ tan ⁡ x 2 d y d x = − c 1 sin ⁡ x + c 2 cos ⁡ x + d d x cos ⁡ x ln ⁡ tan ⁡ x 2 d 2 y d x 2 = − c 1 cos ⁡ x + c 2 sin ⁡ x + d 2 d x 2 cos ⁡ x ln ⁡ tan ⁡ x 2 ⇒ d 2 y d x 2 = − c 1 cos ⁡ x − c 2 sin ⁡ x − cot 2 ⁡ x − 2 − cos ⁡ x ln ⁡ tan ⁡ x 2 ⇒ d 2 y d x 2 + y + cot 2 ⁡ x = 0

2 + c 1 cos ⁡ x + c 2 sin ⁡ x + cos ⁡ x log ⁡ tan ⁡ x 2

cos ⁡ x ⋅ ln ⁡ tan ⁡ x 2 + 2

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The differential equation $\frac{d^{2} x}{d y^{2}} + y + \cot^{2} x = 0$ must be satisfied by

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$2 + c_{1} \cos x + \sqrt{c_{2}} \sin x$

$2 + c_{1} \cos x + c_{2} \sin x + \cos x \log (\tan \frac{x}{2})$

all the above

$\cos x \cdot \ln (\tan \frac{x}{2}) + 2$

answer is B, C.

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Detailed Solution

$(a) : y = 2 + c_{1} \cos x + \sqrt{c_{2}} \sin x \begin{array}{l} \frac{d y}{d x} = - c_{1} \sin x + \sqrt{c_{2}} \cos x \\ \frac{d^{2} y}{d x^{2}} = - c_{1} \cos x - \sqrt{c_{2}} \sin x = 2 - y \\ \frac{d^{2} y}{d x^{2}} + y + 2 = 0 \\ (b) : y = \cos x \ln (\tan \frac{x}{2}) \\ \begin{array}{l} \frac{d y}{d x} = \cos x \frac{\sec^{2} (x / 2)}{2 \tan (x / 2)} - \sin x \ln (\tan \frac{x}{2}) \\ \frac{d y}{d x} = \cot x - \sin x \ln (\tan \frac{x}{2}) \\ \frac{d^{2} y}{d x^{2}} = - \cot^{2} x - 2 - \cos x \ln (\tan \frac{x}{2}) \\ \frac{d^{2} y}{d x^{2}} + y + \cot^{2} x = 0 \end{array} \end{array}$

$(c) : y = 2 + c_{1} \cos x + c_{2} \sin x + \cos x \ln \tan \frac{x}{2} \begin{aligned} \frac{d y}{d x} = - c_{1} \sin x + c_{2} \cos x + \frac{d}{d x} (\cos x \ln (\tan \frac{x}{2})) \end{aligned} \frac{d^{2} y}{d x^{2}} = - (c_{1} \cos x + c_{2} \sin x) + \frac{d^{2}}{d x^{2}} (\cos x \ln (\tan \frac{x}{2})) \Rightarrow \frac{d^{2} y}{d x^{2}} = - c_{1} \cos x - c_{2} \sin x - \cot^{2} x - 2 - \cos x \ln (\tan \frac{x}{2}) \Rightarrow \frac{d^{2} y}{d x^{2}} + y + \cot^{2} x = 0$