I am trying to solve a system of coupled ODEs:
$$ \begin{align} \frac{dn_A}{dt} & = e\left[j(t) - f\, θ_H\sinh\left(\frac{g\,n_A}{T}\right)\right] \\ \frac{dθ_H}{dt} & = a\left[bP\,(1 - θ_H )^2 - c~(θ_H)^2 - f\,θ_H\,\sinh\left(\frac{g\,n_A}{T}\right)\right] \\ \frac{dP}{dt} & =h(i - P) + d\,T\,(P-P_a)\left[bP~(1 - θ_H )^2 - c~(θ_H)^2\right]. \end{align} $$
Here $e, f, g, T, a, b, c,h,i,d$ are constants, $j$ is a time dependent variable, and $P_a$ = 101325. The initial values for the three parameters $n_A, θ_H, P$ are $0, 0.5$, and $8106$ respectively.
I was able to solve the equations using the explicit Euler method, but I am having trouble applying the implicit Euler method to do the same. For the first equation I can use the explicit Euler method to estimate the initial value for $n_A$ at time point $t_{n+1}$, but I am not able to figure out what should I take as a value for the $θ_H$ term that also gets involved in the Newton-Raphson iteration scheme for calculation at $t_{n+1}$.