Harmonic balance Michael Steer derivation?
Whilst the objective of the exercise is to apply an iterative method, based on the Jacobian, this particular system can be solved analytically.

Assuming \$\small e(t) = cos (\omega t)\$, the equation to solve for \$\small v(t)\$ is: $$\small \left ( v(t)\right )^2 +2v(t) - cos (\omega t)=0$$

Solving gives:
$$\small v(t) = -1\: \pm\: \sqrt{1+cos(\omega t)} $$

Decomposing \$\small v(t)\$ into Fourier series components (to do this, express \$\small v(t)\$ in double angle form), the DC component; fundamental; and 2nd harmonic voltages are:

$$\small V_0=-0.1$$
$$\small V_1=0.6$$
$$\small V_2=-0.12$$

which compare with: 

$$\small V_0=-0.09$$
$$\small V_1=0.58$$
$$\small V_2=-0.09$$

for the iterative solution.