Please tell me where I am making an error: I want to use Fourier Transforms to solve the following LDE 1 = (x)n + ( x),,n- 1 + x with lim u(x) = lim u(x) = 0 X - - 00 Taking FT on the DE u" ( x) + u( x) = 1+ x 1 1 1 1 (ik) ' u(x) + u(x) = 1+ x2 = (1 +kux) = 1+ x2 = u(X) : 1+ k2 1 4 2 and the solution will be u(x) = F1( u(x) _ 1 1 First I calculate 1 + 2 , that is e- ikx dx 1+ x2 00 Single poles are x = i and x= -i. e ikx e ikx ek We choose C+ where pole x= i belongs, then, residue 1+ 2 , *=i = lim (x -i) (x- i)(x+1) - = lim x- i (x+i) = 2i Then 1 + x =2 10 2;= nek Now we solve 1 u(X) =- 1 + 2 " e by calculating the inverse FT, 1 ikx 1 = (x)n 1+ knee dk= 1 + 12 - ek(1 + ix) dk 2 1 Poles are k= i and k= -i. We choose C + where pole x= i belongs, then, 1 ek(1 + ix) ek(1 + ix) 1 ek(1tix) ex residue 2 14 2 , k=i =im,2 (k-1) ( k-i(kfi) = im;- k- 12 (kti) = 21 And u(x) = 2 1-,;=nex However I made a mistake since: u"(x) =-ne- And -u"(x) +u(x) =-ne-* +ne-* =0, and I should get 1 1 + x