So your start seem good, but I do have to comment you really can put the x and y in front of the D terms. so we have

Now you want to eliminate x so we multiply the top equation by 4 and let act on the bottom equation to get

Now if we add the two equations we get

This has complimentary solution

Now if we plug this into the 2nd equation we get

This gives

This gives

Now we can answer your question

If you put these into the two equations above you will get what you want