Question

Find the Green’s function ?(?, ?) to solve the boundary value problem:
?" = -12 - ?; ⁡⁡⁡⁡?(0) = 0⁡, ?(2) = 0

3. Find the Green's function G(x, t) to solve the boundary value problem: y" = –12 – y; y(0) = 0, y(2) = 0

Answer 1

TE Given diffeential equation is y" +y = -12, yo so, 702)=0 _0 Corresponding homogeneous dijl eq8 Y"ty =o - ☺ and complementary Solution i yeru ) = C, Sen et Gest Thus, for x4 94,5) = 0,15) Sinxt(219) Sina Cone all have Ç60,5) =0 for osaag, so 9(0,5) = Gos) Boy x = 0 ostos So, 018,5) = 609)Sin e Os xs Second, we have 912, 8) =0, ssasl. So, 9 (2,5) = g(s) sio 2 + (2/8) Gr2 = 0 A solution can be choosen with (2/3) = -6,15) tam 2 This gives Gregs) = C,(9) Sinx-c,(s)tan 2) box This can be simplified by factoring out the Gre) and placing the remaining kamp over a common denominator. 1. The result is 9123) = 4(3)[ Svo X 612 - Sin 2 9mx] 9129) 2001) 1:1) Sim211-2)] En 2 6A 2 L

Love Glxs) = dors) Sim 211-x) SEXEI We note that all could have started with Yz(x)= Sin211-4) as one of the linealy independent Solution of the homogeneous problem in antecipation that Yale) salisfy the second bounday Conclition. Gre,s) = {(1,(s) Sinx oxas [dz(s) Sin 2/1-x) ;q=XEI ç!!!) = { c Sin 211-8) Sinx osass I c Sin 211-x) Seis P ssxal here we choone (rs) = C Sin 211-9), dirs)=( Sin} O GC8T, 1) - (571) =17 Care or oe 414,5) = ? - Sim 201-1) Seis x - Sin2 O < < Sin 2(1-0) Seng Sin 2 ssoal.