So I have done the first part of this question i.e. finding a homogeneous solution and going through the usual steps to get $G(x,s)$ in terms of $x<s$ and $x>s$. My question is for the last part, how does one just write down a particular solution?
2026-03-26 10:58:25.1774522705
How to find particular solution for Green's function with $g(x)$ as Dirac delta function?
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Recall that the Green's function $G(x,s)$ for this BVP solves $$y''-2y'+y = \delta(x-s), \ \ \ y(-1)=y(1)=0.$$ Since this equation is linear, we can add solutions and all that will change is the right-hand side. If we choose $y(x) = G(x,1/2)+G(x,-1/2)$, then $y$ solves the equation with $g(x) = \delta(x-1/2)+\delta(x+1/2)$