Weyl-tensor in GR

Can you derive the Weyl-tensor for the Schwarzschild-solution? What about a gravitational wave? And what about a FLRW-cosmology? Can you think of a physical reason why the expressions work out like that?

Both the Schwarzschild-solution and gravitational waves are vacuum solutions, so the Weyl-tensor is equal to the Riemann-tensor \(C_{\mu\nu\alpha\beta} = R_{\mu\nu\alpha\beta}\), so the direct computation of \(R_{\mu\nu\alpha\beta}\) from the derivatives and the squares of the Christoffel-symbols would be sufficient. FLRW-cosmologies are maximally symmetric spacetimes with no Weyl-curvature at all, \(C_{\mu\nu\alpha\beta} = 0\).