= Solution
Introduce the position and velocity variables $y_1=y$ and $y_2=y'$. Then the required <system of ordinary differential equations> and <initial condition> are
$$
\boxed{\mathbf y'=\mathbf f(x,\mathbf y),\qquad
\mathbf f(x,(y_1,y_2))=\begin{pmatrix}y_2\\F(x,y_1,y_2)\end{pmatrix},\qquad
\mathbf y(a)=\begin{pmatrix}c\\d\end{pmatrix}.}
$$
Thus $f_1(x,\mathbf y)=y_2$. Conversely, the first equation identifies $y_2=y_1'$, and the second recovers the original second-order <ordinary differential equation> and both initial data.
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