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This is an example of the nonuniqueness of solutions.
This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires choosing a gauge.
The constants c and c account for the nonuniqueness of eigenvectors and are not solvable unless an initial condition is given for the system.
Due to the nonuniqueness of eigenvectors, every solution arrived at in this way has undetermined constants c, c, ... c.