To solve the differential equation \(\frac{dy}{dx} = \frac{xy}{1+x^2}\), we first separate the variables:

\(\frac{1}{y} \, dy = \frac{x}{1+x^2} \, dx\).

Integrate both sides:

\(\int \frac{1}{y} \, dy = \int \frac{x}{1+x^2} \, dx\).

This gives:

\(\ln |y| = \frac{1}{2} \ln(1+x^2) + C\).

Exponentiate both sides to solve for \(y\):

\(|y| = e^C \sqrt{1+x^2}\).

Using the initial condition \(y = 2\) when \(x = 0\), we find:

\(2 = e^C \sqrt{1+0^2}\), so \(e^C = 2\).

Thus, \(y = 2\sqrt{1+x^2}\).