A quick way to classify this equation is to look at it feature by feature. Linearity is about how $y$ and $dy/dx$ appear: here both appear to the first power and are never multiplied by each other or by a function of $y$, so the equation is linear in $y$. Homogeneity for a first-order linear equation just means there is no standalone forcing term on the right-hand side after arranging it as $dy/dx + P(x) y = Q(x)$; since the equation is already $dy/dx + x^2 y = 0$ with $Q(x)=0$, it is homogeneous. Exactness is checked differently, by casting the equation as $M\,dx + N\,dy = 0$. Rearranging gives $x^2 y\,dx + dy = 0$, so $M = x^2y$ and $N=1$. The test for exactness is whether $\partial M/\partial y$ equals $\partial N/\partial x$. Computing these, $\partial M/\partial y = x^2$ while $\partial N/\partial x = 0$, and these are not equal for general $x$, so the equation fails the exactness test and is non-exact (even though it is easily solved by separation of variables, that is a different property from exactness). Putting the three properties together: linear, homogeneous, non-exact.
\[\boxed{\text{Linear, homogeneous, and non-exact}}\]