Problem when factoring operators

Question

So I was trying to find a general solution to the second-order linear differential equation: $$\frac{d^2y}{dx^2}+p(x)y=0$$I had read about factoring operators as if they were polynomials. So, I wrote the DE as $$(D^2+p(x))y=0\implies(D+i\sqrt{p(x)})(D-i\sqrt{p(x)})y=0$$So we have that either $$(D+i\sqrt{p(x)})y=0\text{ or }(D-i\sqrt{p(x)})y=0$$Which are first order linear differential equations that are quite trivial to solve. However, the solution to each of these combined is $y=e^{\pm i\int\sqrt{p(x)}dx}$, which is not a solution to the original DE. What did I do wrong here?

Answer 1

The operator $D$ doesn't commute in general: $$p(x)D \ne Dp(x)$$ Because; $$Dp(x)y=p'(x)y+p(x)y'\ne p(x)y'=p(x)Dy$$ But when $p$ is constant we have: $$pD = Dp$$