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  4. The differential equation (d2 y/d x2)+y+ cot 2 x=0 must be satisfied by

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

Differential Equations

Solution:

(1) $\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$
(2) $\frac{d y}{d x}=\cos x \frac{\sec ^2 x / 2}{2 \tan \frac{x}{2}}-\sin x \ln \left(\tan \frac{x}{2}\right) $
$ \frac{d y}{d x}=\cot x-\sin x \ln \left(\tan \frac{x}{2}\right) $
$ \frac{d^2 y}{d x^2}=-\operatorname{cosec}^2 x-\left(\sin x \cdot \frac{1}{\sin x}+\cos x \ln \left(\tan \frac{x}{2}\right)\right)$
$ \frac{d^2 y}{d x^2}=-\cot ^2 x-2-\cos x \ln \left(\tan \frac{x}{2}\right) $
$ \frac{d^2 y}{d x^2}+y+\cot ^2 x=0$
(3) $ \frac{d y}{d x}=\cos x-\sin x$
$ \frac{d^2 y}{d x^2}=-\cos x-\sin x $
$ \frac{d^2 y}{d x^2}+y+\cot ^2 x=\cot ^2 x$