mekkomhada
- 11
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How would I prove that (D-r)^m annihilates x^ke^{rx} where k=0,1,...,m-1.
At first I tried to do it as a computation by expanding the whole the thing out but that turned out to be a nightmare. After that I went down a different path that seems promising but I can't seem to figure it out. Here's what I have so far:
(D-r)(x^ke^{rx})=D(kx^{k}e^{rx})-rkx^{k}e^{rx}<br /> =kx^{k-1}e^{rx}
and
<br /> (D-r)^m=(D-r)^{m-1}(D-r)<br />
I think I'm on the right track here but I'm seriously stuck. I just need a little nudge in the right direction. I appreciate any help.
John
At first I tried to do it as a computation by expanding the whole the thing out but that turned out to be a nightmare. After that I went down a different path that seems promising but I can't seem to figure it out. Here's what I have so far:
(D-r)(x^ke^{rx})=D(kx^{k}e^{rx})-rkx^{k}e^{rx}<br /> =kx^{k-1}e^{rx}
and
<br /> (D-r)^m=(D-r)^{m-1}(D-r)<br />
I think I'm on the right track here but I'm seriously stuck. I just need a little nudge in the right direction. I appreciate any help.
John