This is the PDE, which I have to solve analytically: $$ \frac{\partial G(t, z)}{\partial t}=\sigma\left(1-e^{-z}\right) \frac{\partial G(t, z)}{\partial z}+\sigma\left(e^{-z}-1\right) G(t, z)-\gamma G(t, z)+\gamma $$ $\gamma$ and $\sigma$ are real numbers between 0 and 1.
However something strange happens if I try solving it. The basic method is the method of characteristics, which I tried to implement. Writing down the characteristics,then solving them and combining them into a final solution. However something strange happens if I do that. If I write the characteristics and let Mathematica/Wolfram solve them, then combine them into a final solution I get a different solution than the one wolfram/mathematica provides and I and my professor don't understand why. Can someone help? i'll link the characteristics I wrote below and the solution mathematica provides for the pde: $$ $$ First characteristic $$ \frac{d z}{\sigma\left(e^{-z}-1\right)}=\frac{d G}{\sigma\left(e^{-z}-1\right) G-\gamma G+\gamma} $$ Second characteristic $$\frac{d t}{1}=\frac{d z}{\sigma\left(e^{-z}-1\right)} $$ Solution of the first characteristic: $$ G(z)=c_{1} e^{\left(\gamma \log \left(1-e^{z}\right)+\sigma z\right) / \sigma}+\frac{\gamma\left(1-e^{-z}\right)^{\gamma / \sigma} 2 F_{1}\left(\frac{\gamma+\sigma}{\sigma}, \frac{\gamma+\sigma}{\sigma} ; \frac{\gamma}{\sigma}+2 ; e^{-z}\right)}{\gamma+\sigma} $$ Solution of the second characteristic: $$ c_1=\sigma t+\log(1-e^z) $$ And using the method I've learned:$\phi(c_1,c_2)=0 \rightarrow c_1=h(c_2)$, where h is an arbitrary function and replacing $c_1$,$c_2$ from the equations shown above, I don't get the same solution as wolfram. $$ $$ The wolfram solution: $$ \begin{array}{l}{G(t, z)=\frac{1}{\left(e^{z}-1\right)(\gamma+\sigma)} e^{z-t(\gamma+\sigma)}} \\ {\quad\left(-\gamma e^{\sigma t} c_{1}\left(\sigma+\log \left(e^{z}-1\right)\right)+\gamma e^{\sigma t+z} c_{1}\left(\sigma t+\log \left(e^{z}-1\right)\right)-\sigma e^{\sigma t} c_{1}\left(\sigma t+\log \left(e^{z}-1\right)\right)+\right.} \\ {\left.\sigma e^{\sigma t+z} c_{1}\left(\sigma t+\log \left(e^{z}-1\right)\right)+\gamma e^{t(\gamma+\sigma)}{}_{2} F_{1}\left(1, \frac{\gamma+\sigma}{\sigma} ; \frac{\gamma}{\sigma}+2 ; \frac{1}{1-e^{z}}\right)\right)}\end{array} $$ Where am I mistaken or how to solve it analytically?