I know the method. For a linear homogeneous ordinary differential equation with a root of the characteristic polynomial in $\alpha$ has multiplicity $k$, then $y=x^me^{\alpha x}$ with $m=0,1,\cdots k-1$ is a solution. This however was learnt from need of solving a problem and not in a mathematics course, so I do not know the reason for it.
On reading this book (online edition section 2.1.3), I am bit puzzled at the approach by the author. z''(t)+\Gamma z'(t)+\omega_0^2 z(t)=0 This is readily solved with solutions of the form $z=e^{\alpha t}$ where $\alpha = -\frac{\Gamma}{2} \pm \sqrt{\frac{\Gamma^2}{4}-\omega_0^2}$
Defining $\omega^2\equiv \omega_0^2-\Gamma^2/4$
For the case $\Gamma/2<\omega_0$ we have solutions $z(t) = Ae^{-\Gamma t/2}\cos(\omega t-\theta)\quad -(1)$or, $z(t) =e^{-\Gamma t/2}(c\cos\omega t+d\sin\omega t)\quad -(2)$
For the degenerate case $\Gamma/2=\omega_0$ the author gives the following argument:
.... gives only one solutiopn. One way to find the other solution is to approach the situation from the $\Gamma/2<\omega_0$ case as a limit. Taking $\omega\rightarrow 0$ for (1) gives us $e^{-\Gamma t/2}$ but for (2) we get $0$ However if we divide the second solution by $\omega$ as it does not depend on $t$ we get a non-zero limit $\lim_{\omega\rightarrow 0}\frac{1}{\omega}e^{-\Gamma t/2}\sin\omega t = te^{-\Gamma t/2}$
I have two questiotns here :
- Is this wrong? As the author seems to to have ignored the cosine part of (2) which diverges
- What is the mathemtaical name for this approach as wikipedia does not have any hints