Note 3 Green's function For this problem, the solution u can be written explicitly as (, y)f(y) where G(a, y) is the Green's function given by G(0)={01-) y o show this, we start by recalling that for any function which is twice differ entiable, there are constants CI C2, such that u()=G1+/u'(y)dy C If u satisfies the one dimensional Poisson equation, then C u(a)=C1+C2a f(z)dz dy f(a)dxdy Ly F(y)l F(y)dy F() yf(y)d g (a-y)f(y)dy, by proper attention to dummy variables. Finally, we obtain the general solution❈❊❉✳❋✽●■❍ ❏▲❑✴●☎●✱▼❖◆P▲◗❙❘❚▼✯❯☎❋✶❱❲❉☎▼ ❳❩❨✴❬❪❭❴❫❩❵❜❛❞❝❩❬❡❨✔❢✸❣✖❤✂✐✬❥☎❭❡❫❩❤❦❛❴❨✔❣✙❧♠❭❴❵✙❨✔♥✬♦q♣✪r✔♥s❢✠❤✤t❞❬❴❵✖❭✶❭❡❤✪♥✉❤✚✈♠❝❩❣✙❵❜♣✚❵✖❭❴❣✙✇sr✔❛ ♦②①④③✠⑤✿⑥⑧⑦⑩⑨ ❶❸❷①❹③❁❺❴❻❩⑤✶❼❽①❹❻❩⑤✶❾✔❻ t❞❫❩❤✂❬❴❤ ❷ ①❹③✞❺✶❻❩⑤☛❵❜❛❪❭❴❫❩❤❃❿✢❬❡❤✪❤✂♥✗➀ ❛☛➁❹❧❩♥✸♣✽❭❡❵✖❨✴♥✉➂✔❵✙➃✔❤✪♥✬❢☎✇ ❷ ①❹③❁❺❴❻❩⑤✿⑥➅➄ ❻✳①❲➆❞➇❊③✠⑤ ➈❲❼ ➉▲➊➋❻s➊⑩③ ③②①❲➆❞➇❊❻♠⑤ ➈❲❼ ③■➊❇❻❆➊➌➆ ➍✞❨s❛❴❫❩❨➎t❸❭❴❫❩❵❜❛✂❥✳t➏❤➐❛❲❭❙r✱❬❴❭✥❢☎✇✉❬❴❤✌♣✪r✱❣✙❣✙❵✖♥❩➂❆❭❴❫✸r✱❭★➁❹❨✴❬❖r✔♥☎✇❆➁❹❧❩♥❚♣✽❭❴❵✙❨✔♥❊t❞❫❩❵❜♣❙❫q❵❜❛★❭❲t❞❵❜♣✚❤➐➑♠❵✖➒✠❤✂❬✶➓ ❤✪♥➔❭❡❵✙r✔❢❩❣✖❤✴❥☎❭❴❫❩❤✂❬❴❤❦r✔❬❴❤❖♣✚❨✴♥✸❛✶❭❡r✱♥➔❭❙❛✒→ ⑨ r✔♥✸➑✉→➏➣✴❥✸❛❴❧✸♣❙❫s❭❴❫✸r✱❭ ♦❽①❹③↔⑤↕⑥ → ⑨✯➙ ⑦➋➛ ❶ ♦✠➜➝①❹❻❩⑤❲❾✴❻ ♦➜ ①❹❻❩⑤➞⑥ →➣ ➙ ⑦➋➟ ❶ ♦➜ ➜ ①✁➠➔⑤✶❾✴➠❚➡ ➢➤➁❽♦q❛❴r✱❭❴❵❜❛❲➥✸❤✌❛☛❭❴❫❩❤✤❨✴♥❩❤✤➑♠❵✙✐➐❤✂♥✸❛❴❵✖❨✴♥✸r✱❣✗➦❽❨✴❵✙❛❡❛✶❨✴♥❆❤✌➧✴❧❚r➎❭❴❵✙❨✔♥❁❥☎❭❴❫❩❤✂♥ ♦➜ ①❹❻❩⑤✯⑥➌→➣ ➇ ⑦➋➟ ❶ ❼❽①✁➠➔⑤✶❾✴➠❚➡ ➍❪❫❩❤✂❬❴❤✪➁❹❨✔❬❡❤✔❥ ♦②①④③✠⑤✿⑥✺→ ⑨ ➙ →➣ ③➨➇ ⑦➋➛ ❶➫➩⑦➋➟ ❶ ❼❽①④➠☎⑤❲❾➔➠✔➭✉❾✴❻✠➡ ➯✥❤✚➥✸♥❩❵✙♥❩➂ ➲ ①❹❻❩⑤✯⑥⑧⑦➳➟ ❶ ❼❽①④➠☎⑤❲❾➔➠❚❺ t☛❤✢❨✴❢✸❛❴❤✪❬❡➃✔❤✥❭❴❫✸r✱❭ ⑦➛ ❶ ➩⑦➟ ❶ ❼❽①✁➠➔⑤✶❾✴➠✴➭■❾✔❻ ⑥ ⑦➛ ❶ ➲ ①❹❻❩⑤✶❾✔❻ ⑥ ➵❻➲ ①❹❻❩⑤➤➸ ➛❶ ➇ ⑦➳➛ ❶ ❻➲ ➜➝①❹❻❩⑤❲❾✴❻ ⑥ ③➲ ①❹③✠⑤✯➇ ⑦➋➛ ❶ ❻❩❼❽①❹❻❩⑤✶❾✔❻ ⑥ ⑦➛ ❶ ①④③➨➇➺❻❩⑤❴❼❽①❹❻❩⑤❲❾✴❻↔❺ ❢☎✇❦❝❩❬❡❨✔❝✠❤✪❬✯r✱❭✶❭❴❤✂♥➔❭❴❵✙❨✔♥➐❭❴❨❦➑♠❧✸✐➐✐✎✇❃➃➎r✱❬❡❵❜r✱❢❩❣✙❤✂❛✂➻✞❳②❵✙♥✸r✱❣✙❣✖✇✴❥✔t☛❤❞❨✔❢♠❭❙r✱❵✙♥➐❭❴❫❩❤❞➂✴❤✪♥❩❤✂❬❡r✔❣❩❛✶❨✴❣✖❧♠❭❡❵✖❨✴♥ ❵✙♥s❭❡❫❩❤❖➁❹❨✔❬❡✐ ➼