Using epsilon to rewrite equations in terms of linear combinations

Question

In physics, the damped oscillator is governed by the equation of motion: $\ddot{x} + 2\beta \dot{x}+\omega _{0}^2x=0$ where $\omega _{0}=\beta$ for an oscillator with critical damping. The solution is $$x(t)= e^{-\beta t}\left[C_{1}e^{\sqrt{\beta ^{2}-\omega _{0}^{2}}t}+C_{2}e^{-\sqrt{\beta ^{2}-\omega _{0}^{2}}t}\right]$$ I see in the textbook I use an exercise to finding an alternative solution to the equation of motion and was wondering how to derive this solution by Considering $\beta =\omega _{0}+\epsilon$ and Expanding the solution in $\epsilon$ in order to demonstrate that for small values of $\epsilon/\omega _{0}$. In other words, how can a solution be obtained in which the solution x(t) is rewritten in the form $x(t)\approx e^{-\beta t}[(\tilde{C_{1}})+\tilde{C_{2}}t]$.

Answer 1

I will say up front that I'm primarily a math guy with an interest in physics, but physics is by no means my specialty. That being said, I have done a few problems like this years ago and here is how I would attempt this problem:

If $\frac{\epsilon}{\omega_0}$ is small then $\beta^2 \approx \omega_0^2$ so $\left|\sqrt{\beta ^{2}-\omega _{0}^{2}}\space \right|$ is quite small. For the sake of notation convenience I will define $\sqrt{\beta ^{2}-\omega _{0}^{2}} = \alpha$. Next make a linear approximation with the first two terms of the Taylor series expansion of $e^{\alpha t}$. We have $$e^{\alpha t} \approx 1+\alpha t$$ $$e^{-\alpha t} \approx 1-\alpha t$$ Since $\frac{(\alpha t)^n}{n!}$ becomes arbitrarily small very quickly due to how small $| \alpha |$ is, we have reason to make a linear approximation. Now we have $$\begin{align}x(t) &\approx e^{-\beta t}\left[C_{1}(1+\alpha t)+C_{2}(1-\alpha t)\right] \\ &= e^{-\beta t}\left[C_{1}+C_1 \alpha t+C_{2}-C_2 \alpha t)\right] \\ &= e^{-\beta t}\left[(C_{1}+C_2)+ t(\alpha C_{1}-\alpha C_2) \right] \end{align}$$ Next introduce the new constants $\tilde{C_1} =C_{1}+C_2 $ and $\tilde{C_2} = \alpha C_{1}-\alpha C_2$ to conclude that $$x(t)\approx e^{-\beta t}[(\tilde{C_{1}})+\tilde{C_{2}}t]$$