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(B) The given differential equation is $\cos x(3 \sin x+\cos x+3) dy = (1+y \sin x(3 \sin x+\cos x+3)) dx$.Dividing by $\cos x(3 \sin x+\cos x+3) dx$,

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$2 \log _{e}\left(\frac{2 \sqrt{3}+10}{11}\right)$

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(B) The given differential equation is $\cos x(3 \sin x+\cos x+3) dy = (1+y \sin x(3 \sin x+\cos x+3)) dx$.Dividing by $\cos x(3 \sin x+\cos x+3) dx$,we get:$\frac{dy}{dx} - (	an x)y = \frac{1}{\cos x(3 \sin x+\cos x+3)}$.This is a linear differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$,where $P(x) = -	an x$ and $Q(x) = \frac{1}{\cos x(3 \sin x+\cos x+3)}$.The integrating factor $I.F. = e^{\int -	an x dx} = e^{\ln|\cos x|} = \cos x$ (for $x \in [0, \pi/2)$).The solution is $y \cdot I.F. = \int Q(x) \cdot I.F. dx + C$.$y \cos x = \int \frac{1}{\cos x(3 \sin x+\cos x+3)} \cdot \cos x dx + C = \int \frac{dx}{3 \sin x+\cos x+3} + C$.Using the substitution $t = 	an(x/2)$,$dx = \frac{2 dt}{1+t^2}$,$\sin x = \frac{2t}{1+t^2}$,$\cos x = \frac{1-t^2}{1+t^2}$.$y \cos x = \int \frac{2 dt / (1+t^2)}{3(2t/(1+t^2)) + (1-t^2)/(1+t^2) + 3} + C = \int \frac{2 dt}{6t + 1 - t^2 + 3 + 3t^2} + C = \int \frac{2 dt}{2t^2 + 6t + 4} + C = \int \frac{dt}{t^2 + 3t + 2} + C$.$y \cos x = \int \frac{dt}{(t+1)(t+2)} + C = \int (\frac{1}{t+1} - \frac{1}{t+2}) dt + C = \ln|\frac{t+1}{t+2}| + C$.$y \cos x = \ln|\frac{	an(x/2)+1}{	an(x/2)+2}| + C$.Given $y(0)=0$,$0 = \ln(1/2) + C \Rightarrow C = \ln 2$.$y \cos x = \ln(\frac{1+	an(x/2)}{2+	an(x/2)}) + \ln 2 = \ln(\frac{2(1+	an(x/2))}{2+	an(x/2)})$.For $x = \pi/3$,$	an(x/2) = 	an(\pi/6) = 1/\sqrt{3}$.$y(1/2) = \ln(\frac{2(1+1/\sqrt{3})}{2+1/\sqrt{3}}) = \ln(\frac{2(\sqrt{3}+1)}{\sqrt{3}} \cdot \frac{\sqrt{3}}{2\sqrt{3}+1}) = \ln(\frac{2\sqrt{3}+2}{2\sqrt{3}+1})$.Wait,re-evaluating the integral result: $y \cos x = \ln(\frac{2(1+	an(x/2))}{2+	an(x/2)})$. At $x=\pi/3$,$y(1/2) = \ln(\frac{2(1+1/\sqrt{3})}{2+1/\sqrt{3}}) = \ln(\frac{2(\sqrt{3}+1)}{2\sqrt{3}+1})$.Checking the options,the correct form is $2 \ln(\frac{2\sqrt{3}+10}{11})$.

Let $y=y(x)$ be the solution of the differential equation $\cos x(3 \sin x+\cos x+3) dy = (1+y \sin x(3 \sin x+\cos x+3)) dx$; $0 \leq x \leq \frac{\pi}{2}, y(0)=0$. Then,$y\left(\frac{\pi}{3}\right)$ is equal to ..... .

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