Question

If $y\left(x\right)$ satisfies the differential equation $y^{\prime }-y\tan x=2x\sec x\text{and}y\left(0\right)=0,$ then

• A. $y\left(\frac{\pi }{3}\right)=\frac{{\pi }^2}{9}$
• B. $y\left(\frac{\pi }{4}\right)=\frac{{\pi }^2}{8\sqrt{2}}$
• C. $y^{\prime }\left(\frac{\pi }{3}\right)=\frac{4\pi }{3}+\frac{2{\pi }^2}{3\sqrt{3}}$
• D. $y^{\prime }\left(\frac{\pi }{4}\right)=\frac{{\pi }^2}{18}$

Answer 1

Answer: (B), (C)

$y^{\prime }\left(\frac{\pi }{3}\right)=\frac{4\pi }{3}+\frac{2{\pi }^2}{3\sqrt{3}}$

$\frac{dy}{dx}-y\tan x=2x\sec x$

$\cos x\frac{dy}{dx}-\left(\sin x\right)y=2x$

$d\left(y\cos x\right)=d\left(x^2\right)$

$y\cos x=x^2+c$

$y\left(0\right)=c=0$

$y=x^2\sec x$

also, $y^{\prime }=2x\sec x+x^2\sec x\tan x$