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Proof involving non-autonomous differential equation

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Let A be an n x n matrix. Let f(t; x0) be a solution to the initial value problem
x' = Ax;
x(0) = x0.

Show that f(t2; f(t1; x0)) = f(t1 + t2; x0).
Show by example that this is not true for a non-autonomous differential equation.

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Proof involving non-autonomous differential equation is presented.

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