Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y\frac{dx}{dy}\right)\sin\left(\frac xy\right)$, $y0$ and $x(1)=\frac\pi2$. Then $\cos(x(2))$ is equal to :

Question

Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y\frac{dx}{dy}\right)\sin\left(\frac xy\right)$, $y>0$ and $x(1)=\frac\pi2$. Then $\cos(x(2))$ is equal to :

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] Put $x=vy$: $x-y\frac{dx}{dy}=-y^2\frac{dv}{dy}$, so $y=-y^2v'\sin v$, i.e. $\sin v\,dv=-\frac{dy}y$. Integrating, $\cos v=\ln y+C$. $x(1)=\frac\pi2\Rightarrow v=\frac\pi2$, $C=0$. At $y=2$: $\cos\frac{x(2)}2=\ln2$, so $\cos x(2)=2\cos^2\frac{x(2)}2-1=2(\ln2)^2-1$. Answer (4).

Q. Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y\frac{dx}{dy}\right)\sin\left(\frac xy\right)$, $y>0$ and $x(1)=\frac\pi2$. Then $\cos(x(2))$ is equal to :

Explanation

Reviewed by Dr. Abubacker

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