Question

a) Det6 the green's function for

y"(x)+ y(x) = f(x)

Y(0)=y(1)=0

b) use the result from ( 1) to solve the differential equation

y"(x)+y(x)= x

Subject to the same boundary condition as in (a)

Answer 1

Solution & Green function for YO) = Ya) = 0 Firstly, we shall find green function of the following boundry value problem with = 0, y (.1)=0 – Ceneral solution of 4) is given by yoxy = AxtB – Q die. YO) = 0 B=0 a A+B=0 Yas o - - А=0 There fore equation ② gives only trivial solution There fore green function Celxt) exist for the aucorujated bourday value bonoblow given by ® ] given by saxta osx<t Glit=1 texts six the

the proposed Corean function also satisfy the following property " Cela,f) ů continuous at a=t sie bitt be = aitta (bill)t +63 9g) -0 - - - © (1) The deviretine of a has discontinuity of - I at the point x=t, PE) where Polt) = coefficient of derivative in Ⓡ equal to t the wighest order de contenere Can y tot or het le-t-o lecx, t) must cretisfy the bow dry conditim ® ta Cecost)=0 - 92=0 - 014, ) = 0 = 614b2=0 ☺ Aleo from ② and ③ we have in bz-a = t - - 6

Therefore we have A2=0, B₂ = t , b = -t a = (1-1) putting all these value in ③ me Cockt) = L xlt-t) : osast. Lt(1) test Now comparing &" ty=fers with " + 8 cyao we have x) = 4-fas , so that pe = yes-foty We know that if Cecaith is green's function of associated boundry value problem (giver by (2) and ), then the given boundary value problen can be reduced to following integral eqretin goes - l'acky pes do = ('64[769–ft") ar = ['Gent) yo dt - l'acus. ft dat

Now Ceiven diff. egration y" ty=x put the above fx=x le fest then equation I X Cr-tt yo - Lauren wordt - ['ev. t. de = 'correndo fitoajt dr - Carey = ['con) word – [409)- YOU s'acus.goy dt - $ 01-27) - (4-1)(***) y(x) = f ( x, t) YO) dt - 1 (1 x) - 2 ( x - 3) Ahhur