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Introduce the position and velocity variables $y_1=y$ and $y_2=y^{\prime }$. Then the required system of ordinary differential equations and initial condition areThus $f_1(x,\mathbf{y})=y_2$. Conversely, the first equation identifies $y_2=y_1^{\prime }$, and the second recovers the original second-order ordinary differential equation and both initial data.

Collect the two dependent variables in a column vector. The vector field and initial vector areThe given system of ordinary differential equations is then $Y^{\prime }=F(x,Y)$ with $Y(0)=B$. Both components of $F$ are continuous on all of $\mathbb{R}\times {\mathbb{R}}^2$, since their denominators are at least one.