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Initial-Value Problem for System of Differential Equations : Fundamental Theorem of Calculus

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October 1st, 2022

Let Q(t) =< (less than or equal) C + integral from t_0 to t ( K(s) Q(s) ) ds,
Where Q(t) is a nonegative function , C > 0 and K(s) >= 0.

a).Show that:

Q(t) =< Ce^( integral from t_0 to t ( K(s)ds) ), t >= t_0

b). What conclusion can be made if C = 0? ( Note that proof in a may fail is C = 0 ).

I want a detailed proof, please justify every claim you make. Thanks.

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