Solution
= Solution
For the first-order <linear ordinary differential equation>, an <integrating factor> is
$$
I(x)=\exp\int\frac{4x}{1+x^2}dx=(1+x^2)^2.
$$
Multiplication converts the left side to a <derivative>:
$$
\frac d{dx}\big[(1+x^2)^2y\big]=\frac1{1+x^2}.
$$
Integrating gives $(1+x^2)^2y=\arctan x+C$. The initial value fixes $C=1$, so
$$
\boxed{y(x)=\frac{1+\arctan x}{(1+x^2)^2},\qquad x\in\mathbb R.}
$$