Reducing to first-order differential systems 
Hello, in order to do this, I am aware I need to substitute $x''$ for say $a'$ and $y''$ for say $b'$, but I'm unsure of how this will yield 4 equations?
Furthermore, in terms of wording of the equation, does this mean I do not have to solve it, just find the equations?
Thanks.
 A: An $n^{th}$ order diﬀerential equation can be converted into an $n-$dimensional system  of first order  differential equations.
In your case, each $2^{nd}-$order system reduces to two first-order systems, resulting in four first-order equations.
We have:


*

*$x_1 = x \implies x'_1 = x' = x_2$

*$x_2 = x' \implies x'_2 = x'' = \dfrac{1}{m}f(t, x(t), y(t))$

*$x_3 = y \implies x'_3 = y' = x_4$

*$x_4 = y' \implies x'_4 = y'' = \dfrac{1}{m}g(t, x(t), y(t))$


Recall that you would make the appropriate substitutions into the function $f(t, x(t), y(t))$ and $g(t, x(t), y(t))$ for $t, x_1, x_2, x_3, x_4$ as necessary. 
These are represented as $\tilde x(t), \tilde y(t)$ below to note that there is a change in variables required when you are actually given those functions.
Our reduced system is:
$$\begin{aligned}
x'_1 & = x_2 \\
x'_2 & =  \dfrac{1}{m}f(t, \tilde x(t), \tilde y(t)) \\
x'_3 & = x_4 \\
x'_4 & = \dfrac{1}{m}g(t, \tilde x(t), \tilde y(t)) 
\end{aligned}$$
You cannot solve this as you need the $f(t, x(t), y(t))$ and $g(t(x(t),y(t))$ and would use some numerical method like Euler or Runge-Kutta, for example.
A: Hint: maybe you can simply write
$\begin{cases}m \dot x = a(t, x(t), y(t))\\ 
\dot a = f(t, x(t), y(t))\\
m \dot y = b(t, x(t), y(t))\\ 
\dot b = g(t, x(t), y(t))\end{cases}$
where obviously $a()$ and $b()$ are functions. These are 4 equations. $f$ and $g$ depend on both $x(t)$ and $y(t)$ and you write a unique system because all these equations must be satisfied.
According to the question, it doesn't seem that it is required also the solution. Anyway, because you don't know the shape of $f$ and $g$, you could only copy a general solution from the textbooks.
