Let x=x(y) be the solution of y=(x-ydxdy)xy, y0, x(1)=2.

Question

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

Accepted Answer

$$\frac{dy}{y} = \sin\!\left(\frac{x}{y}\right)d\!\left(-\frac{x}{y}\right) \implies \ln y = \cos\frac{x}{y}+C.$$

Using $x(1)=\pi/2$: $0=\cos(\pi/2)+C \Rightarrow C=0$, so $\ln y=\cos(x/y)$.

At $y=2$: $\cos(x/2)=\ln2$, and $\cos x = 2\cos^2(x/2)-1 = 2(\ln2)^2-1$.

Solutions · 1