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I know for a system to be linear, it has to obey superposition, but how did they go from the combined equation to the 2 equations x(t) = ... and y(t) = ...
Did they solve the ODE and just omit the steps?
edit: does this belong in an electrical engineering stack exchange?
They did not solve the ODE. They are saying that, given two pairs of solutions $(x_1(t),y_1(t))$ and $(x_2(t),y_2(t))$, the linear combination $(k_1x_1(t) + k_2x_2(t), k_1y_1(t) + k_2y_2(t))$ is also a solution to the given ODE. Consequently, the given ODE is linear.