 Solution

ID: past-exam-of-the-algebra-1-and-differential-equations-1-course-of-the-university-of-oxford/2015/6/a/i/solution

Introduce the position and velocity variables

y_{1} = y

and

y_{2} = y^{'}

y^{'} = f (x, y), f (x, (y_{1}, y_{2})) = (y_{2} F (x, y_{1}, y_{2})), y (a) = (c d) .

Thus

f_{1} (x, 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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