Suppose, I would like to solve initial value problem (IVP) $$ (1):\quad \begin{cases} x'(t)=f_1(t,x,y),\\ y'(t)=f_2(t,x,y), \end{cases} $$ with initial conditions $$ (2):\quad x(0)=x_0,y(0)=y_0 $$ and $t\in[t_1,\,t_2]$, but at the same time ''from physics'' is known that there is a constraint $$ (3):\qquad\qquad\left.y'(t)\right|_{t=t_2}=0. $$
Is there any chance to join eqns. $(1)-(3)$ and solve them numerically anyhow?
Update 1
According to request of @Rebellos let me briefly explain motivation and write exact equations.
Exact system was introduced here earlier and may come from more general system in the following form $$ (4):\quad \begin{cases} \partial_x E(x,t) = 1 - n_e(x,t),\\ \partial_x n_e(x,t) = -8E(x,t)n_e(x,t). \end{cases} $$ I considered (4) in every moment of time $\forall t$ as a nonlinear system of ode's in space $x\in[0,\,L]$ $$ (5):\quad \begin{cases} E'(x) = 1 - n_e(x),\\ n'_e(x) = -8E(x)n_e(x). \end{cases} $$ with initial condition $$ (6a):\quad E(0)=E_0,~n_e(0)=n_0. $$ But, ''from physics'' is known that $$ (6b):\quad \partial_x \left.E(x,t)\right|_{x=L}=0. $$
Q: The issue is that how to solve numerically $(4),(6a)$ along with constraint $(6b)$.
Update 2
According to request of @Lutz Lehmann, here constans are $L,n_0,E_0=\text{const},$ space and time variables are $(x,t)\in[0,\,L]\times[0,\,T].$
Remark 1. Maybe (highly likely) that my question is completely incorrent and it has no sence, or maybe (probably not) somebody can classify it as ill-posed problem, for example link.
Remark 2. I thought about the some kind of regularization like
$$
(7):\quad
\begin{cases}
\kappa\cdot\partial_t E(x,t) + \partial_x E(x,t) = 1 - n_e(x,t),\\
\kappa\cdot\partial_t E(x,t) + (n_e(x,t))^{-1}\partial_x n_e(x,t) = -8E(x,t),
\end{cases}
$$
where $\kappa$ is relatively small, let's say $\kappa\approx10^{-4}.$ Adding IC and BC
$$
(8):\quad
\left.E(x,t)\right|_{t=0}\equiv E_0,\,\left.n_e(x,t)\right|_{t=0}\equiv n_0,\,
\left.E(x,t)\right|_{x=0}\equiv E_0,\,\left.n_e(x,t)\right|_{x=0}\equiv n_0,\,
$$
it's not a big deal to solve $(7),(8)$ with Wolfram Mathematica (WM), for example $T=1$, and achieve steady-state solution as follows, and $6b$ is not satisfied in that case.
Update 3
More interesting comparison can be performed. Problem $(5),(6a)$ can be solved in WM directly by classical Runge-Kutta (RK) 2nd and 4th order methods as well as problem $(7),(8),(6b)$ can be solved manually by method of lines (MOL). As a result, the following figure one may get
