Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2015 / 6 / a / i

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.