In this video at around 9:00 , Carl Bender demonstrates a method of solving y''+a(x)y'+b(x)y=0.(adsbygoogle = window.adsbygoogle || []).push({});

He first rewrites it in terms of differential operators

D^{2}+a(x)D+b(x))y(x)=0,

then factors it

(D+A(x))(D+B(x))y=0

then multiplies it out to determine B(x). I thought we would get

(D^{2}+DB+AD+AB)y=0

but at 15:29, he says that D, when it acts on B, either it acts on B or it 'goes past B' and acts on y and because of that, we get two terms, BDandD', so the result is

(D^{2}+BD+B'+AD+AB)y=0

Why doesn't the operator just act on B?

If it only acts on B, then shouldn't BD disappear somehow (and vice-versa)?

Also, this would mean for D to act on something, it has to be the right? (DA≠AD?)

Thanks

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# Multiplying out differential operators

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