Mathematics - Differential Equation Question with Solution | TestHub

$\frac{dy}{dx}+\left(\tan x\right)y=\sin x; 0\le x\le \frac{\pi }{3}$

$I.F.=e^{\int \tan xdx}=e^{\ln \sec x}=\sec x$

$y\sec x=\int \tan xdx+C$

$y\sec x=\ln \left|\sec x\right|+C$

$x=0, y=0 \Rightarrow C=0$

$y\sec x=\ln \left|\sec x\right|$

$y=\cos x·\ln \left|\sec x\right|$

${\overline{)y}}_{x=\frac{\pi }{4}}=\left(\frac{1}{\sqrt{2}}\right)·\ln \sqrt{2}$

${\overline{)y}}_{x=\frac{\pi }{4}}=\frac{1}{2\sqrt{2}}{\log }_{e}2$