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Differential Equations

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The Unique Bounded Solution

DIFFEQ-N1THDE

Suppose $f(x)$ is a continuous, bounded function. There is only one bounded solution for the equation $y'+y=f(x)$ on $x\in (-\infty,\infty)$, which is given by $\int_{-\infty}^xf(s)g(x,s)ds+Ce^{-x}$. (You can use the integrating factor method to find this expression.)

The expression of $g$ and $C$ are

Hint: Use the "carrot" (^) to indicate exponents and use parentheses instead of brackets. Do not use any spaces in your answer.