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Let a function $ \phi : \mathbb{R} \rightarrow \mathbb{R^n}$ that satisfies $|\phi´(t)| \leq L|\phi(t)| $ for some $L$ constant. Show that $|\phi(t)| \leq |\phi(0)| + L\int_{0}^{t} |\phi(t)| dt$ for $t\geq 0$

I think i have to use the Mean value theorem. But its not clear to me how to use it.

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1 Answer 1

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Use the fundamental theorem of calculus and the triangle inequality: $$ \phi(t) = \phi(0) + \int_0^t \phi'(s) \, ds$$ so that $$|\phi(t)| = \left| \phi(0) + \int_0^t \phi'(s) \, ds \right| \le |\phi(0)| + \int_0^t |\phi'(s)| \, ds.$$ Finally apply the hypothesis $|\phi'(s)| \le L |\phi(s)|$.

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