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In your case it would be something like this:

var('t');
x=function('x')(t)
y=function('y')(t)
de1 = diff(x,t) - diff(y,t) == x
de2 = diff(y,t) == y
desolve_system([de1, de2], [x,y])

and the result is as follow:

[x(t) == t*e^t*y(0) + e^t*x(0), y(t) == e^t*y(0)]

In your case it would be something like this:

var('t');
x=function('x')(t)
y=function('y')(t)
de1 = diff(x,t) - diff(y,t) == x
de2 = diff(y,t) == y
desolve_system([de1, de2], [x,y])

and the result is as follow:follows:

[x(t) == t*e^t*y(0) + e^t*x(0), y(t) == e^t*y(0)]