Solving coupled Poisson/Drift Diffusion equations using finite difference method I have a system of coupled non-linear ODEs. 
$$J = \mu*e*n(x)*E(x) + \mu*K*T*\frac{dn(x)}{dx}$$ 
$$\frac{dE(x)}{dx} = \frac{4*\pi*e}{\epsilon}[N_D(x) -n(x)]$$
The first equation is drift-diffusion, the second is Gauss' Law. I am interested in solving this system self-consistently for the carrier profile, $n(x)$, and the electric field, $E(x)$, using finite differences. The current, $J$, the mobility, $\mu$, elementary charge, $e$, are known. The doping profile $N_D(x)$ is known, and I have boundary conditions for the carrier concentration $n(0) = N_D(0)$ and $n(L) = N_D(L)$. I am unsure how to proceed with solving these equations. 
I am following the analysis of Baranger and Wilkins in the attached review. Appendix B. 
Thanks in advance!
 A: You can decouple your system by injecting your equation (2) into
your equation (1). Doing so, you will get a nonlinear second order
differential equation in terms of the electric field $E$. The second 
equation reads as,
$$n = N - \frac{\epsilon}{4\pi e} \dot{E}$$ 
from which you derive,
$$\dot{n} = \dot{N} - \frac{\epsilon}{4\pi e} \ddot{E}$$ 
Letting,
\begin{eqnarray*}
a(x) &=& \frac{4\pi e}{\epsilon}\dot N(x) -\frac{4\pi}{\epsilon \mu kT}J\\   
b(x) &=& \frac{4\pi e^2}{\epsilon kT}N(x)\\
c &=& \frac{e}{kT}
\end{eqnarray*}
The second-order DE can be rewritten,
$$\ddot{E}=a(x) + b(x) E + c E\dot{E}$$
This equation can be casted into the form of Abel's equation of the second kind (of the first derivative order) considering as new dependent variable $E$ and new dependent field $u(E)=\dot E$ so that $\ddot{E}=u\dot u$. It can also 
be casted into the form of Abel's equation of the first kind considering 
the further change of dependent field $v(E)=\frac{1}{u(E)}$. 
Unfortunately I was not able to integrate any of them in closed form. 
