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You want the equation to hold for all values of c0, c1, so you can omit them as variables to solve for:

sage: solve([eq1, eq2], a, b)
[[a == e^(4*c0 + 2*c1), b == 4*c0 + 2*c1]]

To extract the solution for a in particular, you can do e.g.

sage: sols = solve([eq1, eq2], a, b, solution_dict=True); sols
[{a: e^(4*c0 + 2*c1), b: 4*c0 + 2*c1}]
sage: a == sols[0][a]
a == e^(4*c0 + 2*c1)

You want the equation equations to hold for all values of c0, c1, so you can omit them as variables to solve for:

sage: solve([eq1, eq2], a, b)
[[a == e^(4*c0 + 2*c1), b == 4*c0 + 2*c1]]

To extract the solution for a in particular, you can do e.g.

sage: sols = solve([eq1, eq2], a, b, solution_dict=True); sols
[{a: e^(4*c0 + 2*c1), b: 4*c0 + 2*c1}]
sage: a == sols[0][a]
a == e^(4*c0 + 2*c1)