How does Zerilli come up with this solution?

I am studying the paper "Gravitational field of a particle falling in a Scharzschild geometry analyzed in tensor harmonics" by Zerilli. The author calculates the gravitational radiation emitted by a particle falling along a geodesic into a Schwarzschild black hole.

In the appendix I, the author calculates the spectrum of the gravitational wave radiation emitted by a mass falling radially into a Schwarzschild black hole. It seems he is using a Green's function method, but how does he come up with the solution given in eqn. (I10)? Where do the $$\tilde{c}_1$$ and $$\tilde{c}_2$$ come from? It is really unclear.

• He defines $\tilde{c}$ in the following lines after eqn I10. But I haven't looked at the derivation yet. Aug 13, 2020 at 16:26
• I think I have understood the procedure up to equation I5 and equation I10 is clear: what I do not understand is how to match two procedures, how to get the Green functions fron I5. Aug 17, 2020 at 15:46