Given are the following differential equations from the paper by Thorne, Kishino and Felsenstein 1991 (reference):
$ \frac{d f_n(t)}{dt} = \lambda(n-1)f_{n-1}(t)-(\lambda+\mu)nf_n(t)+\mu n f_{n+1}(t) $ with $n>0$
$ \frac{d g_n(t)}{dt} = \lambda(n-1)g_{n-1}(t)-(\lambda+\mu)ng_n(t)+\mu (n+1) g_{n+1}(t) + \mu f_{n+1}(t) $ with $n>0$
$ \frac{d g_0(t)}{dt} = \mu g_1(t)+\mu f_{1}(t)$
$ \frac{d h_n(t)}{dt} = \lambda(n-1)h_{n-1}(t)-(\lambda n +\mu (n-1))h_n(t)+\mu n h_{n+1}(t) $ with $n>0$
Using the following conditions:
$f_0(t) = h_0(t) = 0$
$f_1(0) = h_1(0) = 1$
$f_n(0) = h_n(0) = 0$
$g_n(0) = 0$
The authors derive these solutions to the differential equations (with $n > 0)$ :
$f_n(t) = e^{-\mu t}(1-\lambda\beta(t))(\lambda\beta(t))^{n-1}$
$g_n(t) = (1- e^{-\mu t} -\mu\beta(t))(1-\lambda\beta(t))(\lambda\beta(t))^{n-1}$
$g_0(t) = \mu\beta(t)$
$h_n(t) = (1-\lambda\beta(t))(\lambda\beta(t))^{n-1}$
$\beta(t) = \frac{1-e^(\lambda-\mu)t}{\mu-\lambda e^(\lambda-\mu)t}$
I have two questions regarding these equations:
- Can you give me some hints on how the solutions to the differential equations were obtained?
- Say, you multiply the right side of each of the differential equations by a factor $r$, how would the results change?
Thank you!
Edit: renamed the differential equations for clarrification