Simple Linear Differential Operator Problem

[sriracha]

1. (D+1)(D-x)(2e^x+cosx)



2. None



3. (D+1)(D-x)(2e^x+cos(x))=D(2e^x-sinx-2xe^x-xcosx)+1(2e^x-sinx-2xe^x-xcosx)=-4e^x+2e^x-xcosx-2cosx+xsinx-sinx

The correct answer is 2e^x(xsinx-3sinx+2xcosx)

 

[HallsofIvy]

It is the part that you don't show that is wrong. There is no way that "D"
or "D- 1" can change "2e^x" to "2xe^x". I suggest you do that again.

 

[sriracha]

It is the part that you don't show that is wrong. There is no way that "D"
or "D- 1" can change "2e^x" to "2xe^x". I suggest you do that again.

It's (D-x).

 

[sriracha]

Please open this again as the last poster closed it because he or she misread the equation.

 

[HallsofIvy]

Yes, I misread the equation. No, I did not "close" this thread.
D(2e^x+cos(x))= 2e^x- sin(x)
x(2e^x+cos(x))= 2xe^x+ xcos(x)

(D- x)(2e^x+ cos(x))= 2e^x- 2xe^x- sin(x)- xcos(x)

D(2e^x- 2xe^x- sin(x)- xcos(x))= 2e^x- 2e^x- 2xe^x- cos(x)- cos(x)+ xsin(x)= -2xe^x- 2cos(x)+ xsin(x)

(D+1)(2e^x- 2xe^x- sin(x)- xcos(x))= -2xe^x- 2cos(x)+xsin(x)+ 2e^x- 2xe^x- sin(x)- xcos(x)= 2e^x- 4xe^x- 2cos(x)- sin(x)+ xsin(x)- xcos(x)

What you state as the "correct answer" is impossible. You cannot get e^x multiplied by the trig functions.

 

[sriracha]

Thanks Ivy. This was from the first set of problems on the topic I have worked so I was not sure I was solving it correctly, but looks like I did. The solution in the back of the book must be incorrect. What do the open and closed envelopes next to the threads signify?