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

(a): y=2+c1cos⁡x+c2sin⁡x dydx=−c1sin⁡x+c2cos⁡xd2ydx2=−c1cos⁡x−c2sin⁡x=2−yd2ydx2+y+2=0(b): y=cos⁡xln⁡tan⁡x2dydx=cos⁡xsec2⁡(x/2)2tan⁡(x/2)−sin⁡xln⁡tan⁡x2dydx=cot⁡x−sin⁡xln⁡tan⁡x2d2ydx2=−cot2⁡x−2−cos⁡xln⁡tan⁡x2d2ydx2+y+cot2⁡x=0(c): y=2+c1cos⁡x+c2sin⁡x+cos⁡xln⁡tan⁡x2 dydx=−c1sin⁡x+c2cos⁡x+ddxcos⁡xln⁡tan⁡x2 d2ydx2=−c1cos⁡x+c2sin⁡x+d2dx2cos⁡xln⁡tan⁡x2 ⇒ d2ydx2=−c1cos⁡x−c2sin⁡x−cot2⁡x−2−cos⁡xln⁡tan⁡x2 ⇒ d2ydx2+y+cot2⁡x=0

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

(a): y=2+c1cos⁡x+c2sin⁡x dydx=−c1sin⁡x+c2cos⁡xd2ydx2=−c1cos⁡x−c2sin⁡x=2−yd2ydx2+y+2=0(b): y=cos⁡xln⁡tan⁡x2dydx=cos⁡xsec2⁡(x/2)2tan⁡(x/2)−sin⁡xln⁡tan⁡x2dydx=cot⁡x−sin⁡xln⁡tan⁡x2d2ydx2=−cot2⁡x−2−cos⁡xln⁡tan⁡x2d2ydx2+y+cot2⁡x=0(c): y=2+c1cos⁡x+c2sin⁡x+cos⁡xln⁡tan⁡x2 dydx=−c1sin⁡x+c2cos⁡x+ddxcos⁡xln⁡tan⁡x2 d2ydx2=−c1cos⁡x+c2sin⁡x+d2dx2cos⁡xln⁡tan⁡x2 ⇒ d2ydx2=−c1cos⁡x−c2sin⁡x−cot2⁡x−2−cos⁡xln⁡tan⁡x2 ⇒ d2ydx2+y+cot2⁡x=0

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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

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

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