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Homework Statement
Suppose ##\phi(x)## is a function with a continuous derivative on ##0\leq x<\infty## such that ##\phi'(x)+2\phi(x)\leq 1## for all such ##x## and ##\phi(0)=0##. Show that ##\phi(x)<\frac{1}{2}## for ##x\geq 0##.
The Attempt at a Solution
I tried to solve this like I would any other first order differential equation.
$$
\phi'(x)+2\phi(x)\leq 1\Leftrightarrow e^{2x}(\phi'(x)+2\phi(x))\leq e^{2x}\Leftrightarrow e^{2x}\phi(x)\leq e^{2x}
$$
so
$$
\phi(x)\leq e^{-2x}\int\limits_{x_{0}}^{x}e^{2t}dt + ce^{-2x}=\frac{1}{2}+ce^{-2x}
$$
but that was as far as I could get. Any help would be greatly appreciated.