Let y=y(x) solve (x x),dy+(xy x+y x-1),dx=0, 0

Question

Let $y=y(x)$ solve $(x\cos x)\,dy+(xy\sin x+y\cos x-1)\,dx=0$, $0<x<\dfrac{\pi}{2}$. If $\dfrac{\pi}{3}y\left(\dfrac{\pi}{3}\right)=\sqrt3$, then $\dfrac{\pi}{6}\left[y''\left(\dfrac{\pi}{6}\right)+2y'\left(\dfrac{\pi}{6}\right)\right]$ is \_\_\_\_\_\_.

Answer: 2

Accepted Answer

$$\frac{dy}{dx}+\frac{x\sin x+\cos x}{x\cos x}y = \frac{1}{x\cos x}.$$

I.F. $=x\sec x$. $yx\sec x = \tan x+c$. Using the given condition, $c=\sqrt3$. Differentiating twice and evaluating at $x=\pi/6$ gives the stated combination equal to 2.

Solutions · 1