Consider the problem of predicting the concentration of a reactant in an isothermal batch reactor, where the reaction kinetics is second order (non-linear problem). The reaction rate, r, is given by r = ky², where k is the reaction rate constant (k = 1) and y is the concentration of the reactant. We have

By using the the implicit Euler method, y_n+1 = y_n + h f (x_n+1, y_n+1), and a time step of h = 0.1, we obtain, at the first time step,

As the value y₁ is unknown we need to iterate, e.g. use the Newton method, to find the solution. Let us denote y₁ = z and implement the Newton method to determine the solution at the first time step, t = 0.1. First we reformulate the non-linear equation to F(z) = 0; this gives

The Newton method to find a root of F(z) = 0 and with an initial guess, zi, is given by

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dy dt |y(0) = 1, t = [0, 1]. = f(t, y) = ―r = −y²,