Let $y=y(x)$ be the solution of the differential equation $\left(x^2+4\right)^2 d y+\left(2 x^3 y+8 x y-2\right) d x=0$. If $y(0)=0$, then $y(2)$ is equal to

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Aicharya · Accepted Answer

$\begin{aligned} & \frac{d y}{d x}+y\left(\frac{2 x^3+8 x}{\left(x^2+4ight)^2}ight)=\frac{2}{\left(x^2+4ight)^2} \ & \frac{d y}{d x}+y\left(\frac{2 x}{x^2+4}ight)=\frac{2}{\left(x^2+4ight)^2} \ & 	ext { IF }=e^{\int \frac{2 x}{x^2+4} d x} \ & 	ext { IF }=x^2+4 \ & y 	imes\left(x^2+4ight)=\int \frac{2}{\left(x^2+4ight)^2} 	imes\left(x^2+4ight) \ & y\left(x^2+4ight)=2 \int \frac{d x}{x^2+2^2} \ & y\left(x^2+4ight)=\frac{2}{2} 	an ^{-1}\left(\frac{x}{2}ight)+c \ & 0=0+c=c=0 \ & y\left(x^2+4ight)=	an ^{-1}\left(\frac{x}{2}ight) \ & y 	ext { at } x=2 \ & y(4+4)=	an ^{-1}(1) \ & y(2)=\frac{\pi}{32} \end{aligned}$

Let $y=y(x)$ be the solution of the differential equation $\left(x^2+4\right)^2 d y+\left(2 x^3 y+8 x…

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