Question

If $y=y\left(x\right)$ is the solution of the differential equation $\frac{dy}{dx}+\left(\tan x\right)y=\sin x,0\le x\le \frac{\pi }{3},$ with $y\left(0\right)=0,$ then $y\left(\frac{\pi }{4}\right)$ equal to :

• A. ${\log }_{e}2$
• B. $\frac{1}{2}{\log }_{e}2$
• C. $\left(\frac{1}{2\sqrt{2}}\right){\log }_{e}2$
• D. $\frac{1}{4}{\log }_{e}2$

Answer 1

The correct option is C $\left(\frac{1}{2\sqrt{2}}\right){\log }_{e}2$

I.F. $=e^{\int \tan xdx}$
$=e^{\ln \sec x}=\sec x$
Solution of the equation :
$y\left(\sec x\right)=\int \left(\sin x\right)\left(\sec x\right)dx$
$\Rightarrow \frac{y}{\cos x}=\ln \left(\sec x\right)+c$
Put $x=0,$ we get $c=0$
$\therefore y=\cos x\ln \left(\sec x\right)$
Put $x=\frac{\pi }{4},$
$y=\frac{1}{\sqrt{2}}\ln \sqrt{2}=\frac{1}{2\sqrt{2}}\ln 2$