Question

Determine an appropriate Green’s function either by using the method of variation of parameters or otherwise, for the following boundary value problem: 

-(y''(x)-y(x))=\frac{2}{1+e^{-x}},y(0)=y(1)=0

21 Mar 2023
Answer :
Word Count : 452

To find the Green's function for the given boundary value problem, we can first consider the homogeneous equation:

-(y''(x) - y(x)) = 0

The general solution to this equation is:y_h(x) = c_1e^x + c_2e^{-x}Now, we need to find the Green's function, G(x, ξ), which satisfies:

-(G''(x,ξ) - G(x,ξ)) = δ(x - ξ)

where δ(x - ξ) is the Dirac delta function, which is zero everywhere except at x = ξ where it ___ ____ _____ _____ ______ ______ ___ _______ __________.
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