>>400
y = a x^2 + b x + c

2y = 2a x^2 +2b x + 2c
D y = 2a x + b
(D^2) y = 2a

(D^2-D+2)ｙ = 2a x^2 +(2b-2a)x + 2a-b+2c
= x^2+x

2a = 1
2b-2a = 1
2a-b+2c = 0
より

a = (1/2)
b = 1
c = 0