Question

If $y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\cdots \text{to}\infty }}}$, then $\frac{dy}{dx}=\frac{p\sin x}{q+ry}$, $p,q,r\in N$. Find the minimum positive value of $p+q+r$

Answer 1

To find: Minimum positive value of $p+q+r$

$y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+....}}}$

$\Rightarrow y=\sqrt{\cos x+y}$

$\Rightarrow y^2-y=\cos x$

Differentiating we get

$\Rightarrow 2y{y}^{\prime }-y^{\prime }=-\sin x$

$\Rightarrow y^{\prime }(2y-1)=-\sin x$

$\Rightarrow y^{\prime }=\frac{-\sin x}{2y-1}$

$\Rightarrow \frac{dy}{dx}=\frac{\sin x}{1-2y}$

So $p=1,q=1,r=-2$

So $p+q+r=0$