[Math] Analytical Solution for a Second-Order Nonlinear Differential Equation

ordinary differential equations

In the frame of semiconductor physics, I find myself in front of a smart but difficult second-order nonlinear ODE :
\begin{equation}
\tag{$E$-ODE}
\label{eq:E-ODE}
\boxed{
\phi_T \frac{\mathrm{d}^2 E}{\mathrm{d}{x}^2} + E \frac{\mathrm{d}{E}}{\mathrm{d}{x}} – \frac{e N_D}{\varepsilon} E = \kappa
\text{.}
}
\end{equation}
I prefer to consider general boundary conditons, as I am looking for an general analytical solution.

Equation can be rewritten as

\begin{equation*}
\frac{\mathrm{d}^2 E}{\mathrm{d}{x}^2}
= -\frac{1}{\phi_T} E \frac{\mathrm{d}{E}}{\mathrm{d}{x}} + \frac{e N_D}{\varepsilon \phi_T} E + \frac{\kappa}{\phi_T}
\text{.}
\end{equation*}
Setting
\begin{equation*}
\begin{aligned}
& y \left( x \right) \equiv E \left( x \right) \\
& a \equiv -\frac{1}{\phi_T} \\
& b \equiv \frac{e N_D}{\varepsilon \phi_T} \\
& c \equiv \frac{\kappa}{\phi_T}
\end{aligned}
\end{equation*}
\begin{equation*}
\frac{\mathrm{d}^2 y}{\mathrm{d}{x}^2} = a y \frac{\mathrm{d}{y}}{\mathrm{d}{x}} + b y + c
\end{equation*}
Using shortened notations for derivatives:
\begin{equation}
\tag{$y$-ODE}
\label{eq:y-ODE}
\boxed{
y'' = a y y' + b y + c
\text{.}
}
\end{equation}

Here are my attempts :

  • it is a second-order nonlinear differential equation;
  • it is an autonomous equation : $y'' = F \left( y, y' \right)$;
  • it is a Liénard equation :
    \begin{equation}
    \tag{Liénard}
    \label{eq:Liénard}
    y'' + f \left( y \right) y' + g \left( y \right) = 0
    \text{,}
    \end{equation}
    with $f \left( y \right) = – a y$ and $g \left( y \right) = – b y – c$.
  • with the substitution $w = y'$, it is an Abel equation of the second kind :
    \begin{equation}
    \tag{Abel}
    \label{eq:Abel}
    w w'_{y} + f \left( y \right) w + g \left( y \right) = 0
    \end{equation}

I have tried searching in dedicated textbooks, for instance in Polyanin Handbook of Exact Solutions for Ordinary Differential Equations, but I have got the impression that my ODE has no analytical solution…

What do you think? Does one among you see or know one way to solve this equation?

Under request, I might provide boundary conditions for some specific problem.

Remark: I am familiar with numerical solving of ODE's, but this is not what I am looking for here.

Many thanks forward,

Léopold

Best Answer

$$\frac{d^2y}{dx^2}=ay\frac{dy}{dx}+by+c$$ This is autonomous ODE. One can reduce it to the first order, thanks to the usual change :

$\frac{dy}{dx}=u(y) \quad\to\quad \frac{d^2y}{dx^2}=u\frac{du}{dy}\quad\to\quad u\frac{du}{dy}=a\,y\,u(y)+b\,y+c$

This is an Abel's ODE of the second kind.

The change $\quad u(y)=\frac{1}{v(y)}\quad$ leads to an Abel's ODE of the first kind : $$\frac{dv}{dy}=-(b\,y+c)v^3-a\,y\,v^2$$ For further progress, see :

https://www.hindawi.com/journals/ijmms/2011/387429/#sec2

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