I can't figure out how to solve these under cases he hasn't shown in class and he asked one like this on a quiz. My professor doesn't want us to solve them by multiplying and substituting as shown in the first part of this post which he called ad-hoc, he would like us to use the method involving the Ds which he calls systematic elimination as shown in the second part of the post. I can do it easily when the right side is both zeros but I'm having trouble when there is something like e^(2t) on the right. For example:
x''+x-y''=2e^(-t) > ((D^2)+1)x-(D^2)y=2e^(-t)
x''-x+y''=0 > ((D^2)-1)x-(D^2)x=0
The operational determinate is ((D^2)+1)*(D^2)-((D^2)-1)*(D^2)=2D^4
Thus x=(2e^(-t)*(D^2))-(0*(-D^2))*2D^-4 > x=(2e^(-t)*(D^2))/2D^4 > (2D^4 )x=(2e^(-t)*(D^2))
and y=(((D^2)+1)*0)-(2e^(-t)*((D^2)-1))*2D^-4 > y=(2e^(-t)*((D^2)-1)/2D^4 > (2D^4)y=(2e^(-t)*((D^2)-1)
Normally I've got something like (D^2)(3D+2)x=0 and can solve it with characteristic equations but I'm not sure what to do here.
Thanks in advance for any help!