Linear differential operator / linear transformation

[hholzer]

I have two linear differential operators L_1 = D + 1 and L_2 = D - 2x^2

for L_1(L_2) = (D + 1)(D - 2x^2) = (D)(D - 2x^2) + (1)(D - 2x^2) =
D(D) - D(2x^2) + D - 2x^2 = D^2 + D(1 - 2x^2) - 2x^2

does that look right? I might be making an error somewhere but
my book says:
L_1(L_2) = D^2 + D(1 - 2x^2) - 2x(x + 2)

 

[HallsofIvy]

I have two linear differential operators L_1 = D + 1 and L_2 = D - 2x^2

for L_1(L_2) = (D + 1)(D - 2x^2) = (D)(D - 2x^2) + (1)(D - 2x^2) =
D(D) - D(2x^2) + D - 2x^2 = D^2 + D(1 - 2x^2) - 2x^2

does that look right? I might be making an error somewhere but
my book says:
L_1(L_2) = D^2 + D(1 - 2x^2) - 2x(x + 2)

These are operators- they have to be applied to something. If y is a twice differentiable function then
L_1(L_2(y))= (D+ 1)(Dy- 2x^2y)= D(Dy- 2x^2y)+ 1(Dy- 2x^2y)= (D^2y- 4xy- 2x^2Dy)+ Dy- 2x^2y
= D^2y- (2x^2- 1)Dy- 4xy-2x^2y= (D^2+ (1- 2x^2)D- 2x(x+ 2))y

Remember that you have to use the product rule on D(2x^2y): D(2x^2y)= 2D(x^2)y+ 2x^2Dy= 4xy+ 2x^2Dy.