Solution graphs of linear systems
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Consider the linear system x' = Ax, where A is a n x n real matrix. If x = x(t) is a solution of the linear system, we define:
G(x) = {x(t) : t E R),
which is the set of points on the solution curves x=x(t) in R^n. The set G(t) is called the graph of x = x(t) .
Show that if x(1) = x(1)(t) and x(2) = x(2)(t) are any two solutions of the linear system, then
either the intersection of G(x1) and G(x2) is empty, or G(x1) = G(x2), that is their graphs are either disjointed or the same.
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Solution Summary
In this solution we show that the graphs of two initial value problems involving the same linear system are either disjoint or identical.
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The basic idea of the proof is that the behavior of any linear system satisfying a first-order differential equation is completely determined by a single initial condition. Thus, if the systems x1 and x2 have the same initial condition displaced in time by t2 - t1 , they will exhibit identical behavior at all other times also displaced in time by t2 - ...
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