Question

(1 point) Consider the ordinary differential equation d2G 05 - G = 8(x – xo) on - < x < 0 dr2 where 8 is the delta function. Find the continuous solution G = G(x) such that limz+- G(x) = 0 and limz-40 G(x) = 0. The function u = G(x) is given by G= G for – 0 < x < XO, G = for Xo < x < 0. In your answers, type coas x0.

Answer 1

dz da _G = S(x-xo) 0 for x txo for - 00<x<ko, da - G=0, hence the general solution is of the folm G=A, ex+ Az e for some Constants A,, Az. for xo<x<oo da -a=0, hence the general on us of the form G = B, e²+ B₂ ex for some constants B. , B2 d -x<X<X - SA, e²+ Azex G(x)= xo<xco -Bax Lim G(x)=0 X7-00 Lim G(x)=0 A₂=0) [B,=0 SA, ex G(x) = { By el xo <xo V therefore V x xo< x < oo V B₂ ex G is continuous at x=xo A, exo = B₂ e to B2 = A, XO

2G has finite jump dis continuity of magnitude 1 at X=Xo Lim x=xit E tot x=x+E 1. kim - By EX-E_A, PRO-E] =10 -B₂ e Xo A, exo=1 - 2A, 2X=1 A, = -1 et herefle 66) = f* ex-to for -c0 < x < * 1-I Xo-x for No<x< 0