Why does factorizing linear differential operators work?

Question

When a 2nd-order differential equation is represented as something like: $\big(D^2+(a+b)D+ab\big)(y) = \big((D+a)(D+b)\big)y$.

I don't understand how it can then by solved as: $Du+au$ where $u = (D+b)y$.

I don't understand why this works because when there is a function like: $f(x) = \big(a(x)+b(x)\big)c(x) = \big((a+b)c\big)(x)$, it can't just become $a(u)+b(u)$ where $u = c(x)$. Therefore, why shouldn't this logic apply to the $D$ operator? Does it have something do do with the fundamental difference between operators and functions?

Answer 1

This works because addition and multiplication by a constant and differentiation commute:

$$D(ay+bz)=aDy+bDz.$$

That makes the $D$ operator behave like a polynomial variable.

$$(D+a)y=Dy+ay,$$ $$(D+a)(D+b)y=(D+a)(Dy+by) \\=D(Dy+by)+a(Dy+by) \\=D^2y+Dby+aDy+aby \\=D^2y+bDy+aDy+aby \\=D^2y+(b+a)Dy+aby \\=(D^2+(b+a)D+ab)y.$$

This is not more true when the coefficients are not constant,

$$D(a(x)y)\ne a(x)Dy.$$

Answer 2

An operator is a function as well. Just like how a real-valued function maps a real number to another, an operator maps a real-valued function to another. The differential operator has the following properties

$$D(af(x)) = aD(f(x))$$ $$D(f(x) + g(x)) = D(f(x)) + D(g(x))$$

This means the Differential operator is linear. It is analogous to a linear function. Hence, just how you have polynomial functions as a product of affine variables, you can break down an operator into factors and solve each individually.