Solve this system of equations using elimination for $x(t)$ and $y(t)$

Question

I'm taking an online Differential Equations class and don't understand how to solve this system of equations using elimination.

I tried the typical algebraic method but am running into trouble:

$x'+y'-x=5$

$x+y'+y=1$

So

$y=x'-2x-4$

$x=-5+x'+y'$

But I can't imagine this is how I leave the equation. My guess is the equation should be in some form where I can use separation of variables or something, but I'm not entirely sure.

Any help is greatly appreciated.

Answer 1

Even more basic than the good solution from Mhenni Benghorbal to whom all credit must be given.

Your system being $$x'+y'-x=5\tag 1$$ $$x+y'+y=1\tag 2$$ From $(2)$, extract $x$ $$x=1-y-y'\tag 3$$ Differentiate to get $$x'=-y'-y''\tag 4$$ Plug $x$ from $(3)$ and $x'$ from $(4)$ into $(1)$ to get $$-y'-y''+y'-1+y+y'=5\tag 5$$ that is to say $$-y''+y'+y=6 \tag 6$$ Solve $(6)$ for $y$ and plug the result in $(3)$ to get $x$ directly.

Answer 2

Just write the system as

$$(D-1)x+Dy=5\\ x+(D+1)y=1$$

where $D=\frac{d} {dx} $. Applying the differential operator $- (D-1)$ to the second equation and adding both equations gives the equation

$$ -(D^2-1)y+Dy=6 \implies y''-y'-y=6. $$

I think you can finish it!