# Inequality of ODE solutions

Says I have two (scalar) ODE: $u' = f(u,t)$ and $v' = g(v,t)$ where

• Both $f$ and $g$ are piecewise-continuous and locally Lipschitz, for existence & uniqueness of solutions $u(t)$ and $v(t)$.
• $f(x,t) \leq g(x,t)$ for all $x$ and $t$.

I believe that if $u(0) \leq v(0)$ then $u(t) \leq v(t)$ for all $t \geq 0$. But I don't know if there is such theorem, or if not, how to prove it.

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This link can be interesting for you: en.wikipedia.org/wiki/Gronwall's_inequality – Ilya Sep 15 '11 at 20:28