Help Setting up an Equation to use the Elimination Method

[daedie]

Homework Statement

The question is to solve the IVP: x'=2x+y-e^2t & y'=x+2y, where x(0)=1, y(0)=-1

Homework Equations

Arranging the equations and substituting in D for the derivatives, the equations become:
1. (D-2)x-Dy= -2e^2t
2. (D-2)y-Dx= 0

The Attempt at a Solution

My first attempt was to eliminate the y variable and leave x to solve for. But, looking at the problem, I'm having an issue with figuring out how to set the equation up in order to do so. One attempt was to eliminate the (D-2)x & y on both equations:

(D-2)x-Dy=-2e^2t *(D-2)y
(D-2)y-Dx=0 *(D-2)x

(D-2)x(D-2)y-Dy(D-2)y=(D-2)-2e^2t
(D-2)x(D-2)y-Dx(D-2)x=0

Subtracting, this leaves:

Dx(D-2)x-Dy(D-2)y=0

This is far messier than we've dealt with in class, but not beyond the realm something the teacher might give us. I'm wondering if there's an easier way to clear out one of the terms in order to make the Diff Eq easy to solve for x. Once I get one value solved, I can go back and figure out the other. Just the set up is tricky. Thanks for any help you can provide!

EDIT: I was able to get some help. Multiply (1) by D and (2) by (D-2) and voila!

 

[CompuChip]

Note that you wrote "y'=x+2y" as the second equation, and then proceeded to work with "2. (D-2)y-Dx= 0".
So somehow you have introduced a derivative operator on the x?