Saturday, 13 May 2017

calculus - Why can this differential equation be written in 3 different ways?



Suppose we have the following differential equation using operator notation: (Dx)(D+x)y=0 where D=ddx




Now I could rewrite (1) as (Dx)(D+x)y=(ddxx)(y+xy)=y+(xy)xyx2y=y+y+xyxyx2yyx2y+y=0



Or, by switching the order of the brackets; I could rewrite (1) as (Dx)(D + x)y=(D+x)(D - x)y=(ddx+x)(yxy)=y(xy)+xyx2y=yyxy+xyx2yyx2yy=0



Lastly, I could rewrite (1) as (Dx)(D+x)y=(D2x2)y=(d2dx2x2)y=yx2y yx2y=0



There's no doubt there's most probably a simple explanation for it; but how can the same differential equation (1) be written in three different ways: (a), (b), (c)?



Many thanks.


Answer




D+x and Dx do not commute. Your (b) is a demonstration of this.



In (c) you are equating (D+x)(Dx) and D2x2, but that is wrong for a similar reason - in D2x2 the first D is no longer acting on the second x.


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