Let $y$ be a non-zero quadratic polynomial satisfying the differential equation
\[
\left(2+x^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-k y=0,
\]
where $k$ is a real constant. If $y(1)=1$, then the value of the integral
\[
\int_{0}^{1} 2 y d x
\]
is
- $\frac{1}{3}$
- $\frac{2}{3}$
- $1$
- $\frac{4}{3}$
