Question

If $\cos x\frac{dy}{dx}+y\sin x=1$ and $y\left(\frac{\pi }{4}\right)=\sqrt{2}$, then $y\left(\frac{-\pi }{3}\right)$ is

• A. $\frac{1-\sqrt{3}}{2}$
• B. $\frac{1+\sqrt{3}}{2}$
• C. $1-\sqrt{3}$
• D. $1+\sqrt{3}$

Answer 1

The correct option is A $\frac{1-\sqrt{3}}{2}$

$\cos x\frac{dy}{dx}+y\sin x=1\Rightarrow \frac{dy}{dx}+y\tan x=\sec x$
which is a linear differential equation.
$\text{I.F}.=e^{\int \tan xdx}=\sec x$

The general solution is
$y\times \sec x=\int {\sec }^{2}xdx\Rightarrow y\sec x=\tan x+c$
Given $y\left(\frac{\pi }{4}\right)=\sqrt{2}$
$\Rightarrow 2=1+c\Rightarrow c=1$
Hence, the solution of the given equation is $y\sec x=\tan x+1$

Putting $x=\frac{-\pi }{3}$, we get
$2y=-\sqrt{3}+1\Rightarrow y=\frac{1-\sqrt{3}}{2}$