Successive approximations for second order differential equations

Ho can we obtain the (approximate) solution for, a $2$nd order differential equation, using successive approximations?

All the info I found so far explains how to do it for a 1st order equation. Can you point me to the general method for a $2$nd order case?

Transform the second order equation $y''=f(x,y,y')$ in a first order system, introducing a new unknown $z=y'$: \begin{align} y'&=z\\ z'&=f(x,y,z)\\ y(x_0)&=y_0\\ z(x_0)&=z_0 \end{align} The successive iterations are defined by \begin{align} y_{n+1}&=y_0+\int_{x_0}^xz_n(t)\,dt\\ z_{n+1}&=z_0+\int_{x_0}^xf(t,y_n(t),z_n(t))\,dt \end{align}