If y=y(x) solves dydx+( x)y= x, 0 x/3, y(0)=0, then y(4) is:

Question

If $y=y(x)$ solves $\dfrac{dy}{dx}+(	an x)y=\sin x$, $0\le x\le\pi/3$, $y(0)=0$, then $y\left(\dfrac{\pi}{4}ight)$ is:

Accepted Answer

I.F.$=\sec x$. $y\sec x = \log(\sec x)+c$; using $y(0)=0 \Rightarrow c=0$.

$$y=\cos x\log(\sec x) \implies y\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt2}\log(\sqrt2) = \frac{1}{2\sqrt2}\log2.$$

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