Question

(16 pts) Given boundary value problem (1 - 2)y" + 2xy' = 1 y(0) = 0, y'(1) = 0 (a) (6 pts) yı = 1 is a solution to homogeneous equation (1 – 22)y" + 2xy' = 0, find a second solution y2 by reduction of order method. (b) (6 pts) Find Green's function G (1, t) of the BVP. (c) (4 pts) Find a solution of the BVP using G (2,t).

Answer 1

Given BUP is, (;-xxx) 5'+zx y'=1 or y" + + 2xy y'= 900) = 0, 4'() -0. Then (a general sogh of the cinnittyet 2x = 0 is, 4") = 4 tea) + G g(x), where fra) = 4 = 1 and by reduction of order method. -SP(m) are fla) = f(x) s da, P(a) = 2x ( 21 de = J e [frus] - dre . loj (2²1) J (22_1) die F(-x) te, i.second Solution is. Yn (E3) (6) new, soon of ® is, 97) = G+-a) :: yrra) = 6 (X2 HO) = 0 = G=0. Then g(n) = (2 3 2 - x) and off w(2, 42) = 1 $ ) =(2:1) 0 Then G(*.4) ,04% 9, (+) 42(2) W(+) y, Y2(+) w() 7572

Oztur t2-1 (t) MLt21 3 :22-1 serth of the orp is, dt E² 92) = 5(,t)f (t) dt (fet)='} . Gr(2,4)= 962) = ) at + = () (en # +43 (t'+31+) + ² / (+²-3 het 7 dt 4cm) (3-5) 21(**) * ={{462-1*23cm + 2in(n-1)} 6+2142) -(+(+y un (141)} t²-1 2-1 2 ht! 2 2 - 2/2 (251) +2 ln(n+1)