If y=y(x), y(0,2), solves ydydx-(x+y)-(x-y)=0, y(0)=0, then 5y'(2) is ______.

Question

If $y=y(x)$, $y\in\left(0,\dfrac{\pi}{2}\right)$, solves $\sec y\dfrac{dy}{dx}-\sin(x+y)-\sin(x-y)=0$, $y(0)=0$, then $5y'\left(\dfrac{\pi}{2}\right)$ is \_\_\_\_\_\_.

Answer: 2

Accepted Answer

$\sec y\dfrac{dy}{dx}=2\sin x\cos y \implies \sec^2y\,dy = 2\sin x\,dx \implies \tan y=-2\cos x+c$.

$y(0)=0 \Rightarrow c=2$. At $x=\pi/2$: $\tan y=2 \Rightarrow \sec^2y\cdot y'=2\sin x=2 \Rightarrow y'=\dfrac25$.

$$5y'\left(\frac{\pi}{2}\right)=2.$$

Solutions · 1