# Use the method of elimination to find a ODE satisfied by $x(t)$ and $y(t)$

$y''+x'-x+y = \sin t$

$x''+y'-x+y = e^t$

Any ideas?

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Subtract the second equation from the first equation to get: $(y-x)''-(y-x)'=sin t-e^t$ Let z=y-x Solve this to get z then substitute y=x+z in one of the equations you gave.