The problem you have described is known as the Green's function problem, where we are looking for an operator G(x,s) that satisfies the equation LG(x,s) = H(x-s), where H(x) is the Heaviside step function. The inverse problem is then to find the operator L given a function G(x,s).
To answer your question, the functions a(x), b(x), and c(x) are not directly related to the values of G(x,s) and its derivatives. However, they may be used in the process of solving for G(x,s) and L. The function G(x,s) itself is a solution to the differential equation LG(x,s) = H(x-s), and its derivatives may be used to find the operator L. The functions a(x), b(x), and c(x) may be involved in the process of solving for G(x,s) and determining L, but they do not directly determine the values of G(x,s) and its derivatives.
In general, the relationship between G(x,s) and its derivatives and the functions a(x), b(x), and c(x) will depend on the specific problem and the method used to solve it. However, these functions may be related through the differential equation and the boundary conditions of the problem. Ultimately, the values of G(x,s) and its derivatives will depend on the specific problem at hand and the chosen approach to solving it.
In summary, the functions a(x), b(x), and c(x) are not directly related to the values of G(x,s) and its derivatives, but they may be involved in the process of finding these values and determining the operator L. The specific relationship between these functions will depend on the problem and the solution method used.
