Answer
$\displaystyle y + \frac13y^3 = \frac13x^3 + C$
Work Step by Step
First separate the variables:
\begin{align*}
\frac{dy}{dx} &= \frac{x^2}{1+y^2}\\[0.3cm]
(1+y^2) \, dy &= x^2 \, dx
\end{align*}
Now integrate both sides:
\begin{align*}
\int(1+y^2) \, dy &= \int x^2 \, dx\\[0.3cm]
y + \frac13y^3 &= \frac13x^3 + C
\end{align*}
Note that there isn't a way to cleanly isolate $y$ by itself on one side. So our answer must be an implicit function.
