Solve the Following Questions

$\cos x(3 \sin x+\cos x+3) d y$

$=(1+y \sin x(3 \sin x+\cos x+3)) d x$

$\frac{d y}{d x}-(\tan x) y=\frac{1}{(3 \sin x+\cos x+3) \cos x}$

I.F. $=e^{\int-\tan x d x}=e^{f n \cos x \mid}=|\cos x|$

$=\cos x \forall x \in\left[0, \frac{\pi}{2}\right)$

Solution of D.E.

$y(\cos x)=\int(\cos x) \cdot \frac{1}{\cos x(3 \sin x+\cos x+3)} d x+C$

$y(\cos x)=\int \frac{d x}{3 \sin x+\cos x+3} d x+C$

$y(\cos x)=\int \frac{\left(\sec ^{2} \frac{x}{2}\right)}{2 \tan ^{2} \frac{x}{2}+6 \tan \frac{x}{2}+4} d x+C$

Now

Let $\mathrm{I}_{1}=\int \frac{\left(\sec ^{2} \frac{\mathrm{x}}{2}\right)}{2\left(\tan ^{2} \frac{\mathrm{x}}{2}+3 \tan \frac{\mathrm{x}}{2}+2\right)} \mathrm{dx}+\mathrm{C}$

Put $\tan \frac{x}{2}=t \Rightarrow \frac{1}{2} \sec ^{2} \frac{x}{2} d x=d t$

$I_{1}=\int \frac{d t}{t^{3}+3 t+2}=\int \frac{d t}{(t+2)(t+1)}$

$=\int\left(\frac{1}{t+1}-\frac{1}{t+2}\right) d t$

$=\ell \mathrm{n}\left|\left(\frac{\mathrm{t}+1}{\mathrm{t}+2}\right)\right|=\ell \mathrm{n}\left|\left(\frac{\tan \frac{\mathrm{x}}{2}+1}{\tan \frac{\mathrm{x}}{2}+2}\right)\right|$

So solution of D.E.

$\mathrm{y}(\cos \mathrm{x})=\ell \mathrm{n}\left|\frac{1+\tan \frac{\mathrm{x}}{2}}{2+\tan \frac{\mathrm{x}}{2}}\right|+\mathrm{C}$

$\Rightarrow y(\cos x)=\ell n\left(\frac{1+\tan \frac{x}{2}}{2+\tan \frac{x}{2}}\right)+C \quad$ for $0 \leq x<\frac{\pi}{2}$

Now, it is given $\mathrm{y}(0)=0$

$\Rightarrow 0=\ell \mathrm{n}\left(\frac{1}{2}\right)+\mathrm{C} \quad \Rightarrow \quad \mathrm{C}=\ell \mathrm{n} 2$

$\Rightarrow y(\cos x)=\ell n\left(\frac{1+\tan \frac{x}{2}}{2+\tan \frac{x}{2}}\right)+\ell n 2$

For $x=\frac{\pi}{3}$

$y\left(\frac{1}{2}\right)=\ell n\left(\frac{1+\frac{1}{\sqrt{3}}}{2+\frac{1}{\sqrt{3}}}\right)+\ell \mathrm{n} 2$

$\mathrm{y}=2 \ell \mathrm{n}\left(\frac{2 \sqrt{3}+10}{11}\right)$