Here are my system of equations:
$$x'+y'-x=5$$
$$x'+y'+y=1$$
I rearranged them like so:
$$x=x'+y'-5$$
$$y=1-x'-y'$$
I took the derivative of
$$x=x'+y'-5$$
and got
$$x'=x''+y''\Rightarrow y''=x'-x''$$
This is where everything just went south for me...
Since I don't have a $y''$ anywhere I took the second derivative of the easiest equation involving $y$ that I could find:
$$y'=1-x'-y'$$
$$y''=-x''-y''$$
Which means:
$$y''=-x''-(x'-x'')\Rightarrow y''=-x''-x'+x''\Rightarrow y''=-x'$$
But from here I'm just completely confused and don't know where and what I should be substituting anymore. I've tried googling "elimination for differential equation" and I get nothing. It's an online class and there is no documentation on how to use elimination for system of equations with multiple variables and derivatives.
Any help is appreciated.
One has $$5+x=x'+y'=1-y\ ,$$ which implies $x+y=-4$, hence $x'+y'=0$. This then leads to the single constant solution $x=-5$, $y=1$.