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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, $t\geq 0$

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    @user No, in the specific context, the author refers to the equilibrium position as the as equilibrium position of the undamped harmonic oscillator.2011-08-15

3 Answers 3

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Set it equal to zero and then solve

$A_+ e^{-\Gamma_+ t} +A_- e^{-\Gamma_- t}=0$

$ e^{(\Gamma_+-\Gamma_-)t}=- A_+/A_- $

The right side must be positive for there to be a real solution at all, so we'll assume that. Note also how $\Gamma_+-\Gamma_-$ can be written. Then

$t=\frac{\ln(-A_+/A_-)}{2\sqrt{(\Gamma/2)^2-\omega_0^2}}$

is the unique solution, because the exponential function has a single-valued inverse when considered strictly over real numbers.

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I start with:

$0 = Ae^{-Bt}+Ce^{-Dt}$

Then we have $Ae^{-Bt} = -Ce^{-Dt}$. Suppose WLOG that $B\geq D$. Then we have $\dfrac{Ae^{-Bt}}{e^{-Dt}} = Ae^{-(B+D)t} = -C$

Can you prove that a generic exponential function is injective? (this would prevent there from being more than one solution - but it is NOT surjective, so there is no promise that there is a solution).

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    @Christian: Yes, that is the idea. But will you remove your comment? I intended this answer as a hint, and therefore incomplete.2011-08-15
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You mean the question is to prove that $x(t) = 0$ has only one solution. And actually, there is at most one solution.

If $A_+$ and $A_-$ have the same sign, then so do both terms in the definition of $x(t)$, and there is no solution for $x(t) = 0$.

If $A_+$ and $A_-$ have opposite signs, then you want to solve $\lvert A_+ \rvert e^{-\Gamma_+t} = \lvert A_- \rvert e^{-\Gamma_-t}$. Take logs of both sides; you get a linear equation in $t$, and you're done.

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    Then there is the case $A_+ = A_- = 0$. Note the word "crosses" in the question...2011-08-15