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I see what you want to do, but apparently solve will not work with two equations and two derivatives in this way. I think this must have something to do with how Maxima works or how Sage is passing info to Maxima.

Another option would be to use x1 and y1 in place of your derivatives and then solve, since you are really not using the fact that you have functions of t in this solving procedure.

var('x y x1 y1')
de1 = x1 - y1 == x
de2 = y1 == y
sol = solve([de1,de2],[x1,y1])
sol

This gives:

[[x1 == x + y, y1 == y]]

Another option would be to use x1 and y1 in place of your derivatives and then solve, since you are really not using the fact that you have functions of t in this solving procedure.

var('x y x1 y1')
de1 = x1 - y1 == x
de2 = y1 == y
sol = solve([de1,de2],[x1,y1])
sol

This gives:

[[x1 == x + y, y1 == y]]

To find the Jacobian, you need the appropriate vector field. You can get this from the solution with the following

F=map(lambda q: q.rhs(),sol[0])

Then, the Jacobian is:

jacobian(F,[x,y])

Does that resolve the issue?