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  4. The differential equation satisfied by the function y = √ sin x + √ sin x + √ sin x + .....∞ is

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Q. The differential equation satisfied by the function $y = \sqrt{\sin \, x + \sqrt{\sin \, x + \sqrt{\sin \, x + .....\infty}}}$ is

Continuity and Differentiability

Solution:

$y = \sqrt{\sin \, x + \sqrt{\sin \, x + \sqrt{\sin \, x + .....\infty}}}$
$\Rightarrow \:\: y = \sqrt{sin \, x + y} \Rightarrow \, y^2 = \sin \, x + y $
On differentiating both sides, we get
$2y \frac{dy}{dx} =\cos x+ \frac{dt}{dx} \Rightarrow \frac{dy}{dx} \left(2y -1\right)=\cos x $