Differential equation help?
I know that that factors of a linear differential operator with constant coefficients commute.
Suppose that L is a linear operator that can be factored but has variable coefficients.
For example, let L=x(D-1).
Do the factors of L commute?
Provide evidence to support your answer.
Hint: If L=PQ, then L(f(x))=P(Q(f(x)), that is, PQ denotes the compostion of the two operators P and Q.
[x(D-1)]y = x ((D-1)y) = x (dy/dx – y) = x dy/dx – xy.
[(D-1)x]y = (D-1)(xy) = (1y + x dy/dx) – y = x dy/dx.
Hence, linear differential operators with variables don’t generally commute.
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