Question

The solution of $\frac{dy}{dx}+2y\tan x=\sin x$, given that $y=0,x=\frac{\pi }{3}$ is

• A. $y=\cos x-2{\cos }^{2}x$
• B. $y=-\cos x+2{\cos }^{2}x$
• C. $y=\cos x-{\cos }^{2}x$
• D. $y=2\cos x-2{\cos }^{2}x$

Answer 1

The correct option is B $y=-\cos x+2{\cos }^{2}x$

$\frac{dy}{dx}+2y\tan x=\sin x$

Solving linear differential equation,

$\Rightarrow IF=e^{\int 2\tan xdx}$

$\Rightarrow y{e}^{\int 2\tan xdx}=\int \sin x{e}^{\int 2\tan x}dx⟶\left(1\right)$

If $e^{\int 2\tan xdx}$

$=e^{2ln\left|\sec x\right|}$

$=e^{ln{\sec }^{2}x}$

If ${\sec }^{2}x$

putting value in $\left(1\right)$

$\Rightarrow y{\sec }^{2}x=\int \sin x{\sec }^{2}xdx$

$=\int \frac{\sin x}{{\cos }^{2}x}dx$

$\Rightarrow y{\sec }^{2}x=\frac{-1}{\cos x}+c$

$y=0$ $,$ $x=\frac{\pi }{3}$

$\Rightarrow 0=-2+C$

$C=2$

So, $y{\sec }^{2}x=-\frac{1}{\cos x}+2$

$y=-\cos x+2{\cos }^{2}x.$

Hence, solved.