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## Homework Statement

Given w'' - w = f(x)

w'(0) = 1

w'(1) = 0

## Homework Equations

Find the Green's Function

## The Attempt at a Solution

The solution to the homogeneous equation is known as:

w(x) = A*exp(-x) + B*exp(x)

For G's function we have:

u(x) = A1*exp(-x) + B*exp(x), u'(0) = 1

v(x) = A2*exp(-x) + B*exp(x), v'(1) = 0

The homogeneous equation is easily solved by plugging in two B.C.s. However, with Green's function we have two equations with two unknowns and one B.C. each. This leaves:

u'(0) = 1 -> B1 = 1 + A1 -> u(x) = A1*exp(-x) + (1+A1)*exp(x)

v'(1) = 0 -> B2 = A2*exp(-2) -> v(x) = A2*exp(-x) + A2*exp(-2)*exp(x)

Since G1 = A(c)*u(x) and G2 = B(c)*v(x), and we have two jump conditions at point c, I don't see how this can be solved. I've done other problems where A(c) and B(c) can be merged with A1 and A2, but the "1+A1" term keeps me from reducing this two two unknowns and two equations. Am I missing something? Is there an implied BC that I'm not seeing?

Thanks