Consider L(G) = (xx0) with L = d dx p d dx +q subject to the boundary conditions G(0
Chapter 9, Problem 9.3.15(choose chapter or problem)
Consider L(G) = (xx0) with L = d dx p d dx +q subject to the boundary conditions G(0, x0) = 0 and G(L, x0) = 0. Introduce for all x two homogeneous solutions, y1 and y2, such that each solves one of the homogeneous boundary conditions: L(y1)=0, L(y2)=0 y1(0) = 0, y2(L)=0 dy1 dx (0) = 1, dy2 dx (L)=1. Even if y1 and y2 cannot be explicitly obtained, they can be easily calculated numerically on a computer as two initial value problems. Any homogeneous solution must be a linear combination of the two.*(a) Solve for G(x, x0) in terms of y1(x) and y2(x). You may assume that y1(x) = cy2(x). (b) What goes wrong if y1(x) = cy2(x) for all x and why?
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