Hi, I'm stuck understanding a paper by David G Kendall (Stochastic Processes and Population Growth, 1949). The paper demonstrates how to derive $p_n(t) = \mathbb{P}(X_t = n)$ for a pure birth process X, where the birth rate is a constant $\lambda$. In words, we are finding the probability we are state n after some time t.
He starts off by finding the Kolmogorov Forward Equations, and derives $$\frac{d}{dt}p_n(t) = (n-1)\lambda p_{n-1}(t) - n\lambda p_n(t) \\ \text{where } \frac{d}{dt}p_1(t) = -n\lambda p_1(t)$$
Then he puts this into a generating function of the form $$\phi(z,t) = \sum_{n=0}^{\infty}z^np_n(t) \\ \implies \frac{d}{dt}\phi(z,t) = \sum_{n=0}^{\infty}z^n\frac{d}{dt}p_n(t)\\ \iff \frac{d}{dt}\phi(z,t) = \sum_{n=0}^{\infty}z^n(n-1)\lambda p_{n-1}(t) - n\lambda p_n(t)$$
Eventually he gets the following result, he does this by manipulating the sums.
$$\frac{d}{dt}\phi(z,t) = \lambda z(z-1)\frac{d}{dz}\phi(z,t)$$
I understand how he gets to the equation above, but I'm struggling to figure out how he does the following steps, I'll copy his steps exactly.
"We must satisfy the partial differential equation $\frac{d}{dt}\phi(z,t) = \lambda z(z-1) \frac{d}{dz} \phi(z,t)$
The most general solution to which is of the form: $$\phi(z,t) = \Phi\{(1-\frac{1}{Z})e^{\lambda t}\}\\ \text{But } \phi(z,0) = \Phi(1-\frac{1}{Z}) = Z \\ \text{and so } \Phi(Z) = \frac{1}{1-Z} \\ \text{and } \phi(z,t) = \frac{ze^{-\lambda t}}{1-z(1-e^{-\lambda t})}"$$
He then takes expansion in powers of z, to get $p_n(t) = e^{-\lambda t}(1-e^{-\lambda t})^{n-1}$.
Could someone teach me the techniques he uses to identify the most general solution to the partial differential equation $\frac{d}{dt}\phi(z,t) = \lambda z(z-1) \frac{d}{dz}\phi(z,t)$. My knowledge of partial differential equations is limited.
Sorry for the long post, thanks for any help.
Let's do something equivalent to the method of characteristics, but a bit more intuitive. In analogy with the equation $\partial_t\phi=\partial_z\phi$, which has as general solution $\phi(z,t)=F(z+t)$, where $F$ is an arbitrary function, we try to solve $$ \partial_t\phi=\lambda z(z-1)\partial_z\phi\tag{1} $$ with the ansatz $\phi(z,t)=F(g(z)+t)$. Substituting it in $(1)$, we verify that it indeed is a solution, provided $$ 1=\lambda z(z-1)g'(z)\Rightarrow g(z)=\frac{1}{\lambda}\ln\left(1-\frac{1}{z}\right)+C, \tag{2} $$ hence $$ \phi(z,t)=F\left(\frac{1}{\lambda}\ln\left(1-\frac{1}{z}\right)+C+t\right) =F\left(\frac{1}{\lambda}\ln\left(C'\left(1-\frac{1}{z}\right)e^{\lambda t}\right)\right). \tag{3} $$ Since the composition of an arbitrary function with a known function is also an arbitrary function, we may rewrite $(3)$ as $$ \phi(z,t)=\Phi\left(\left(1-\frac{1}{z}\right)e^{\lambda t}\right). \tag{4} $$