Consider an equation of the form $x(t) = A_+ e^{-\Gamma_+ \; t}+A_- e^{-\Gamma_{-}\; t}$
$A_{\pm},\Gamma_{\pm}$ are real constants, both $\Gamma_+$ and $\Gamma_-$ are greater than zero and $t\geq 0$.
Question is to prove that it has only one solution. I tried taking the derivative and equating to zero, but couldn't see any way to prove.
Additional Note: The above question is abstracted from a physical problem. The expression I wrote for $x(t)$ denotes displacement as a function of time, and this particular form is for an "overdamped" oscillator. The actual question asks that "Prove that an overdamped oscillator crosses the equilibriam position only once." Where $x=0$ is the equilibrium position.
$A_{\pm},\Gamma_{\pm}$ are real constants. $t$ is real, and as it is time, $t\geq 0$