MHB Proving the Uniqueness of Solutions to Autonomous Systems: A Study on λ and R_n

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i am studying autonomous system, i came across this proposition
λ is in R and vector b is in R_n where the IVP

x' = Ax + c , with x(λ) = b has at most one solution

is it possible to prove this
 
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It is possible to prove this. What tools are available to you? For example, are you allowed to use the standard existence and uniqueness theorem where the hypotheses involve Lipschitz functions?
 

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